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The study of '''geodesics on an ellipsoid''' arose in connection with geodesy | |||
specifically with the solution of [[triangulation network]]s. The | |||
[[figure of the earth]] is well approximated by an | |||
''[[oblate ellipsoid]]'', a slightly flattened sphere. A ''[[geodesic]]'' | |||
is the shortest path between two points on a curved surface, i.e., the analogue | |||
of a [[straight line]] on a plane surface. The solution of a triangulation | |||
network on an ellipsoid is therefore a set of exercises in spheroidal | |||
trigonometry {{harv|Euler|1755}}. | |||
[[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|150px|[[Isaac Newton]]]] | |||
[[File:Leonhard Euler.jpg|thumb|150px|[[Leonhard Euler]]]] | |||
If the earth is treated as a [[sphere]], the geodesics are | |||
[[great circles]] (all of which are closed) and the problems reduce to | |||
ones in [[spherical trigonometry]]. However, {{harvtxt|Newton|1687}} | |||
showed that the effect of the rotation of the earth results in its | |||
resembling a slightly oblate ellipsoid and, in this case, the | |||
[[equator]] and the [[meridian (geography)|meridians]] are the only | |||
closed geodesics. Furthermore, the shortest path between two points on | |||
the equator does not necessarily run along the equator. Finally, if the | |||
ellipsoid is further perturbed to become a [[triaxial ellipsoid]] (with | |||
three distinct semi-axes), then only three geodesics are closed and one | |||
of these is unstable. | |||
The problems in geodesy are usually reduced to two main cases: the | |||
''direct problem'', given a starting point and an initial heading, find | |||
the position after traveling a certain distance along the geodesic; and | |||
the ''inverse problem'', given two points on the ellipsoid find the | |||
connecting geodesic and hence the shortest distance between them. | |||
Because the flattening of the earth is small, the geodesic distance | |||
between two points on the earth is well approximated by the great-circle | |||
distance using the | |||
[[Earth radius#Mean radius|mean earth radius]]—the relative error is | |||
less than 1%. However, the course of the geodesic can differ | |||
dramatically from that of the great circle. As an extreme example, | |||
consider two points on the equator with a longitude difference of | |||
179°59′; while the connecting great circle follows the | |||
equator, the shortest geodesics pass within | |||
180 km of either [[geographical pole|pole]] (the | |||
flattening makes two symmetric paths passing close to the poles shorter | |||
than the route along the equator). | |||
Aside from their use in geodesy and related fields such as navigation, | |||
terrestrial geodesics arise in the study of the propagation of signals | |||
which are confined (approximately) to the surface of the earth, for | |||
example, sound waves in the ocean {{harv|Munk|Forbes|1989}} and the | |||
radio signals from lightning {{harv|Casper|Bent|1991}}. Geodesics are | |||
used to define some [[maritime boundaries]], which in turn determine the | |||
allocation of valuable resources as such | |||
[[mineral rights|oil and mineral rights]]. Ellipsoidal geodesics also | |||
arise in other applications; for example, the propagation of radio waves | |||
along the fuselage of an aircraft, which can be roughly modeled as a | |||
[[prolate spheroid|prolate (elongated)]] ellipsoid | |||
{{harv|Kim|Burnside|1986}}. | |||
Geodesics are an important intrinsic characteristic of curved surfaces. | |||
The sequence of progressively more complex surfaces, the sphere, an | |||
[[ellipsoid of revolution]], and a triaxial ellipsoid, provide a useful | |||
family of surfaces for investigating the general theory of surfaces. | |||
Indeed Gauss's work on the | |||
[[:de:Gaußsche Landesaufnahme|survey of Hanover]], which involved | |||
geodesics on an oblate ellipsoid, was a key motivation for his | |||
[[Differential geometry of surfaces|study of surfaces]] | |||
{{harv|Gauss|1828}}. Similarly the existence of three closed geodesics | |||
on a triaxial ellipsoid turns out to be a general property of | |||
[[closed surface|closed]], [[simply connected]] surfaces; this was | |||
conjectured by {{harvtxt|Poincaré|1905}} and proved by | |||
{{harvtxt|Lyusternik|Schnirelmann|1929}} | |||
{{harv|Klingenberg|1982|loc=§3.7}}. | |||
== Geodesics on an ellipsoid of revolution == | |||
There are several ways of defining geodesics | |||
{{harv|Hilbert|Cohn-Vossen|1952|pp=220–221}}. A simple definition | |||
is as the shortest path between two points on a surface. However it is | |||
frequently more useful to define them as paths with zero | |||
[[geodesic curvature]]—i.e., the analogue of [[straight lines]] on a | |||
curved surface. This definition encompasses geodesics traveling so far | |||
across the ellipsoid's surface (somewhat less than half the | |||
circumference) that other distinct routes require less distance. | |||
Locally, these geodesics are still identical to the shortest distance | |||
between two points. | |||
By the end of the 18th century, an ellipsoid of revolution (the term | |||
[[spheroid]] is also used) was a well-accepted approximation to the | |||
[[figure of the Earth]]. The adjustment of [[triangulation network]]s | |||
entailed reducing all the measurements to a [[reference ellipsoid]] and | |||
solving the resulting two-dimensional problem as an exercise in | |||
spheroidal trigonometry {{harv|Bomford|1952|loc=Chap. 3}}. | |||
[[File:Geodesic problem on an ellipsoid.svg|thumb|right| | |||
Fig. 1. | |||
A geodesic ''AB'' on an ellipsoid of revolution. ''N'' is the north | |||
pole and ''EFH'' lie on the equator.]] | |||
It is possible to reduce the various geodesic problems into one of two | |||
types. Consider two points: ''A'' at latitude | |||
φ<sub>1</sub> and longitude λ<sub>1</sub> and | |||
''B'' at latitude φ<sub>2</sub> and longitude | |||
λ<sub>2</sub> (see Fig. 1). The connecting geodesic | |||
(from ''A'' to ''B'') is ''AB'', of length | |||
''s''<sub>12</sub>, which has [[azimuth]]s α<sub>1</sub> and | |||
α<sub>2</sub> at the two endpoints.{{refn| | |||
Here α<sub>2</sub> is the ''forward'' azimuth at ''B''. | |||
Some authors calculate the ''back'' azimuth instead; this is given by | |||
α<sub>2</sub> ± π.}} The two geodesic problems usually | |||
considered are: | |||
# the ''direct geodesic problem'' or ''first geodesic problem'', given ''A'', α<sub>1</sub>, and ''s''<sub>12</sub>, determine ''B'' and α<sub>2</sub>; | |||
# the ''inverse geodesic problem'' or ''second geodesic problem'', given ''A'' and ''B'', determine ''s''<sub>12</sub>, α<sub>1</sub>, and α<sub>2</sub>. | |||
As can be seen from Fig. 1, these problems involve solving the triangle | |||
''NAB'' given one angle, α<sub>1</sub> for the direct | |||
problem and λ<sub>12</sub> = λ<sub>2</sub> − λ<sub>1</sub> for the | |||
inverse problem, and its two adjacent sides. | |||
In the course of the 18th century these problems were elevated | |||
(especially in literature in the German language) to the | |||
[[:de:geodätische Hauptaufgabe|''principal geodesic problems'']] | |||
{{harv|Hansen|1865| | |||
p=[http://books.google.com/books?id=WlsOAAAAYAAJ&pg=PA69 69]}}. | |||
For a sphere the solutions to these problems are simple exercises in | |||
[[spherical trigonometry]], whose solution is given by | |||
[[Solution of triangles#Two sides and the included angle given|formulas | |||
for solving a spherical triangle]]. | |||
(See the article on [[great-circle navigation]].) | |||
[[File:Alexis Clairault.jpg|thumb|150px|[[Alexis Clairault]]]] | |||
[[File:Barnaba Oriani.jpg|thumb|150px|[[Barnaba Oriani]]]] | |||
For an ellipsoid of revolution, the characteristic constant defining the | |||
geodesic was found by {{harvtxt|Clairaut|1735}}. A | |||
systematic solution for the paths of geodesics was given by | |||
{{harvtxt|Legendre|1806}} and | |||
{{harvtxt|Oriani|1806}} (and subsequent papers in | |||
[[#{{harvid|Oriani|1808|}}|1808]] and | |||
[[#{{harvid|Oriani|1810|}}|1810]]). | |||
The full solution for the direct problem (complete with computational | |||
tables and a worked out example) is given by {{harvtxt|Bessel|1825}}.{{refn| | |||
This prompted a courteous note by {{harvtxt|Oriani|1826}} noting his | |||
previous work, of which, presumably, Bessel was unaware, and also a | |||
thinly veiled accusation of [[plagiarism]] from {{harvtxt|Ivory|1826}} | |||
(his phrase was "second-hand from Germany"), which prompted an angry | |||
rebuttal by {{harvtxt|Bessel|1827}}.}} | |||
Much of the early work on these problems was carried out by | |||
mathematicians—for example, [[Adrien-Marie Legendre|Legendre]], | |||
[[Friedrich Bessel|Bessel]], and [[Carl Friedrich Gauss|Gauss]]—who | |||
were also heavily involved in the practical aspects of [[surveying]]. | |||
Beginning in about 1830, the disciplines diverged: those with an | |||
interest in geodesy concentrated on the practical aspects such as | |||
approximations suitable for field work, while mathematicians pursued the | |||
solution of geodesics on a triaxial ellipsoid, the analysis of the | |||
stability of closed geodesics, etc. | |||
During the 18th century geodesics were typically referred to as "shortest | |||
lines".{{refn| | |||
{{harvtxt|Clairaut|1735}} uses the [[circumlocution]] "perpendiculars to | |||
the meridian"; this refers to Cassini's proposed map projection for | |||
France {{harv|Cassini|1735}} where one of the coordinates was the | |||
distance from the Paris meridian.}} | |||
The term "geodesic line" was coined by {{harvtxt|Laplace|1799b}}: | |||
<blockquote> | |||
Nous désignerons cette ligne sous le nom de ''ligne géodésique'' [We | |||
will call this line the ''geodesic line'']. | |||
</blockquote> | |||
This terminology was introduced into English either as "geodesic line" | |||
or as "geodetic line", for example {{harv|Hutton|1811}}, | |||
<blockquote> | |||
A line traced in the manner we have now been describing, or deduced from | |||
trigonometrical measures, by the means we have indicated, is called | |||
a ''geodetic'' or ''geodesic line:'' it has the property of being | |||
the shortest which can be drawn between its two extremities on the | |||
surface of the earth; and it is therefore the proper itinerary | |||
measure of the distance between those two points. | |||
</blockquote> | |||
In its adoption by other fields "geodesic line", frequently shortened, | |||
to "geodesic", was preferred.{{refn| | |||
{{harvtxt|Kummell|1883}} attempted to introduce the word "brachisthode" | |||
for geodesic. This effort failed.}} | |||
This section treats the problem on an ellipsoid of revolution (both | |||
oblate and prolate). The problem on a triaxial ellipsoid is covered in | |||
the next section. | |||
=== Equations for a geodesic === | |||
[[File:Friedrich Wilhelm Bessel (1839 painting).jpg|thumb|150px|[[Friedrich Bessel]]]] | |||
{{multiple image | |||
|align=right | |||
|direction=horizontal | |||
|width=220 | |||
|image1=Differential element of a meridian ellipse.svg | |||
|caption1=Fig. 2. Differential element of a meridian ellipse. | |||
|image2=Differential element of a geodesic on an ellipsoid.svg | |||
|caption2=Fig. 3. Differential element of a geodesic on an ellipsoid. | |||
}} | |||
Here the equations for a geodesic are developed; these allow the | |||
geodesics of any length to be computed accurately. The following | |||
derivation closely follows that of {{harvtxt|Bessel|1825}}. | |||
{{harvtxt|Bagratuni|1962|loc=§15}}, | |||
{{harvtxt|Krakiwsky|Thomson|1974|loc=§4}}, and | |||
{{harvtxt|Rapp|1993|loc=§1.2}} also provide derivations of these | |||
equations. | |||
Consider an ellipsoid of revolution with equatorial radius | |||
''a'' and polar semi-axis ''b''. Define the | |||
flattening ''f'' = (''a'' − ''b'')/''a'', eccentricity | |||
''e''<sup>2</sup> = ''f''(2 − ''f'') second eccentricity | |||
''e''′ = ''e''/(1 − ''f''). (In most applications in geodesy, the | |||
ellipsoid is taken to be oblate, ''a'' > ''b''; however, the theory | |||
applies without change to prolate ellipsoids, ''a'' < ''b'', in | |||
which case ''f'', ''e''<sup>2</sup>, and ''e''′<sup>2</sup> are | |||
negative.) | |||
Let an elementary segment of a path on the ellipsoid have length | |||
''ds''. From Figs. 2 and 3, we | |||
see that if its azimuth is α, then ''ds'' can | |||
is related to ''d''φ and ''d''λ by | |||
:(1)<math>{\color{white}.}\qquad | |||
\cos\alpha\,ds = \rho\,d\phi = - dR/\sin\phi, \quad | |||
\sin\alpha\,ds = R\,d\lambda,</math> | |||
where ρ is the | |||
[[Earth radius#Meridional|meridional radius of curvature]], | |||
''R'' = ν cosφ is the radius of the circle of latitude | |||
φ, and ν is the | |||
[[Earth radius#Normal|normal radius of curvature]]. | |||
The elementary segment can therefore be expressed as | |||
:<math>\begin{align}ds &= \sqrt{\rho^2\phi'^2 + R^2}\,d\lambda \\ | |||
&\equiv L(\phi,\phi')\,d\lambda, | |||
\end{align} | |||
</math> | |||
where φ′ = ''d''φ/''d''λ and ''L'' depends on | |||
φ through ρ(φ) and | |||
''R''(φ). The length of an arbitrary path between | |||
(φ<sub>1</sub>, λ<sub>1</sub>) and (φ<sub>2</sub>, λ<sub>2</sub>) is | |||
given by | |||
:<math> s_{12} = | |||
\int_{\lambda_1}^{\lambda_2} L(\phi, \phi')\,d\lambda,</math> | |||
where φ is a function of λ satisfying | |||
φ(λ<sub>1</sub>) = φ<sub>1</sub> and | |||
φ(λ<sub>2</sub>) = φ<sub>2</sub>. The shortest path or geodesic | |||
entails finding that function φ(λ) which minimizes | |||
''s''<sub>12</sub>. This is an exercise in the | |||
[[calculus of variations]] and the minimizing condition is given by the | |||
[[Beltrami identity]], | |||
:<math>L - \phi' \frac{\partial L}{\partial \phi'} = \text{const.} | |||
</math> | |||
[[File:Construction for parametric latitude.svg|thumb| | |||
Fig. 4. | |||
The geometric construction for parametric latitude, β. A point | |||
''P'' at latitude φ on the meridian (red) is mapped to a point | |||
''P′'' on a sphere of radius ''a'' (shown as a blue circle) by | |||
keeping the radius ''R'' constant.]] | |||
Substituting for ''L'' and using Eq. (1) gives | |||
:<math>R\sin\alpha = \text{const.}</math> | |||
{{harvtxt|Clairaut|1735}} first found this [[Clairaut's relation|relation]], | |||
using a geometrical construction.{{refn| | |||
{{harvtxt|Laplace|1799a}} showed that a particle constrained to move on | |||
a surface but otherwise subject to no forces moves along a geodesic for | |||
that surface. Thus, Clairaut's relation is just a consequence of | |||
[[conservation of angular momentum]] for a particle on a surface of | |||
revolution. A similar proof is given by | |||
{{harvtxt|Bomford|1952|loc=§8.06}}.}} Differentiating this | |||
relation and manipulating the result gives | |||
{{harv|Jekeli|2012|loc=Eq. (2.95)}} | |||
:<math>d\alpha=\sin\phi\,d\lambda.</math> | |||
This, together with Eqs. (1), leads to a system of | |||
[[ordinary differential equations]] for a geodesic | |||
:(2)<math>{\color{white}.}\qquad\displaystyle | |||
\frac{d\phi}{ds} = \frac{\cos\alpha}{\rho};\quad | |||
\frac{d\lambda}{ds} = \frac{\sin\alpha}{\nu\cos\phi};\quad | |||
\frac{d\alpha}{ds} = \frac{\tan\phi\sin\alpha}{\nu}.</math> | |||
We can express ''R'' in terms of the | |||
[[Latitude#Reduced (or parametric) latitude|parametric latitude]], | |||
β, using | |||
:<math>R = a\cos\beta</math> | |||
(see Fig. 4 for the geometrical construction), and Clairaut's | |||
relation then becomes | |||
:<math>\sin\alpha_1\cos\beta_1 = \sin\alpha_2\cos\beta_2.</math> | |||
{{multiple image | |||
|align=right | |||
|direction=horizontal | |||
|width=220 | |||
|image1=Geodesic problem on a sphere.svg | |||
|caption1=Fig. 5. Geodesic problem mapped to the auxiliary sphere. | |||
|image2=Geodesic problem mapped to the auxiliary sphere.svg | |||
|caption2=Fig. 6. The elementary geodesic problem on the auxiliary sphere. | |||
}} | |||
This is the [[Spherical trigonometry#Sine rules|sine rule]] of spherical | |||
trigonometry relating two sides of the triangle ''NAB'' (see | |||
Fig. 5), ''NA'' = ½π − β<sub>1</sub>, and | |||
''NB'' = ½π − β<sub>2</sub> and their opposite angles | |||
''B'' = π − α<sub>2</sub> and ''A'' = α<sub>1</sub>. | |||
In order to find the relation for the third side | |||
''AB'' = σ<sub>12</sub>, the ''spherical arc length'', and included | |||
angle ''N'' = ω<sub>12</sub>, the ''spherical longitude'', it is | |||
useful to consider the triangle ''NEP'' representing a geodesic | |||
starting at the equator; see Fig. 6. In this figure, the | |||
variables referred to the auxiliary sphere are shown with the | |||
corresponding quantities for the ellipsoid shown in parentheses. | |||
Quantities without subscripts refer to the arbitrary point | |||
''P''; ''E'', the point at which the geodesic crosses | |||
the equator in the northward direction, is used as the origin for | |||
σ, ''s'' and ω. | |||
[[File:Differential element of a geodesic on a sphere.svg|thumb| | |||
Fig. 7. | |||
Differential element of a geodesic on a sphere.]] | |||
If the side ''EP'' is extended by | |||
moving ''P'' infinitesimally (see Fig. 7), we | |||
obtain | |||
:(3)<math>{\color{white}.}\qquad | |||
\cos\alpha\,d\sigma = d\beta, \quad | |||
\sin\alpha\,d\sigma = \cos\beta\,d\omega.</math> | |||
Combining Eqs. (1) and (3) gives differential | |||
equations for ''s'' and λ | |||
:<math>\frac1a\frac{ds}{d\sigma} | |||
= \frac{d\lambda}{d\omega} | |||
= \frac{\sin\beta}{\sin\phi}.</math> | |||
Up to this point, we have not made use of the specific equations for an | |||
ellipsoid, and indeed the derivation applies to an arbitrary surface of | |||
revolution.{{refn|It may be useful to impose the restriction that the | |||
surface have a positive curvature everywhere so that the latitude be | |||
single valued function of ''Z''.}} | |||
Bessel now specializes to an ellipsoid in which | |||
''R'' and ''Z'' are related by | |||
:<math>\frac{R^2}{a^2} + \frac{Z^2}{b^2} = 1,</math> | |||
where ''Z'' is the height above the equator (see Fig. 4). | |||
Differentiating this and setting | |||
''dR''/''dZ'' = −sinφ/cosφ gives | |||
:<math>\frac{R\sin\phi}{a^2} - \frac{Z\cos\phi}{b^2} = 0;</math> | |||
eliminating ''Z'' from these equations, we obtain | |||
:<math>\frac Ra = \cos\beta = \frac{\cos\phi}{\sqrt{1-e^2\sin^2\phi}}.</math> | |||
This relation between β and φ can be | |||
written as | |||
:<math>\tan\beta = \sqrt{1-e^2} \tan\phi = (1-f) \tan\phi,</math> | |||
which is the normal definition of the | |||
[[Latitude#Reduced (or parametric) latitude|parametric latitude]] | |||
on an ellipsoid. Furthermore, we have | |||
:<math>\frac{\sin\beta}{\sin\phi} = \sqrt{1-e^2\cos^2\beta},</math> | |||
so that the differential equations for the geodesic become | |||
:<math>\frac1a\frac{ds}{d\sigma} = \frac{d\lambda}{d\omega} | |||
= \sqrt{1-e^2\cos^2\beta}.</math> | |||
The last step is to use σ as the independent | |||
parameter{{refn| Other choices of independent parameter are possible. | |||
In particular many authors use the vertex of a geodesic (the point of | |||
maximum latitude) as the origin for σ.}} in both of | |||
these differential equations and thereby to express ''s'' and | |||
λ as integrals. Applying the sine rule to the vertices | |||
''E'' and ''G'' in the spherical triangle | |||
''EGP'' in Fig. 6 gives | |||
:<math>\sin\beta = \sin\beta(\sigma;\alpha_0) = | |||
\cos\alpha_0 \sin\sigma,</math> | |||
where α<sub>0</sub> is the azimuth at ''E''. | |||
Substituting this into the equation for ''ds''/''d''σ and | |||
integrating the result gives | |||
:(4)<math>{\color{white}.}\qquad | |||
\begin{align} | |||
\frac sb &= \int_0^\sigma | |||
\frac{\sqrt{1 - e^2 \cos^2\beta(\sigma';\alpha_0)}}{1-f}\,d\sigma'\\ | |||
&= \int_0^\sigma \sqrt{1 + k^2 \sin^2\sigma'}\,d\sigma', | |||
\end{align} | |||
</math> | |||
where | |||
:<math>k = e'\cos\alpha_0,</math> | |||
and the limits on the integral are chosen so that | |||
''s''(σ = 0) = 0. {{harvtxt|Legendre|1811|p=180}} pointed out | |||
that the equation for ''s'' is the same as the equation for the | |||
[[Meridian arc#Meridian distance on the ellipsoid|arc on an ellipse]] | |||
with semi-axes ''a''(1 − ''e''<sup>2</sup> sin<sup>2</sup>α<sub>0</sub>)<sup>1/2</sup> and | |||
''b''. In order to express the equation for | |||
λ in terms of σ, we write | |||
:<math>d\omega = \frac{\sin\alpha_0}{\cos^2\beta}\,d\sigma,</math> | |||
which follows from Eq. (2) and Clairaut's relation. | |||
This yields | |||
:(5)<math>{\color{white}.}\qquad | |||
\begin{align} | |||
\lambda - \lambda_0 &= (1-f) \sin\alpha_0 | |||
\int_0^\sigma\frac | |||
{\sqrt{1 + k^2\sin^2\sigma'}} | |||
{1 - \cos^2\alpha_0\sin^2\sigma'}\,d\sigma'\\ | |||
&= \omega - \sin\alpha_0 | |||
\int_0^\sigma\frac | |||
{e^2}{1 + \sqrt{1 - e^2\cos^2\beta(\sigma';\alpha_0)}}\,d\sigma'\\ | |||
&= \omega - f\sin\alpha_0 | |||
\int_0^\sigma\frac | |||
{2-f}{1 + (1-f)\sqrt{1 + k^2\sin^2\sigma'}} | |||
\,d\sigma', | |||
\end{align}</math> | |||
and the limits on the integrals are chosen | |||
so that λ = λ<sub>0</sub> at the equator crossing, | |||
σ = 0. | |||
In using these integral relations, we allow σ to | |||
increase continuously (not restricting it to a range | |||
[−π, π], for example) as the great circle, | |||
resp. geodesic, encircles the auxiliary sphere, resp. ellipsoid. The | |||
quantities ω, λ, and ''s'' | |||
are likewise allowed to increase without limit. Once the problem is | |||
solved, λ can be reduced to the conventional range. | |||
This completes the solution of the path of a geodesic using the | |||
auxiliary sphere. By this device a great circle can be mapped exactly | |||
to a geodesic on an ellipsoid of revolution. However, because the | |||
equations for ''s'' and λ in terms of the | |||
spherical quantities depend on α<sub>0</sub>, the mapping is not | |||
a consistent mapping of the surface of the sphere to the ellipsoid or | |||
vice versa; instead, it should be viewed merely as a convenient tool for | |||
solving for a particular geodesic. | |||
There are also several ways of approximating geodesics on an ellipsoid | |||
which usually apply for sufficiently short lines | |||
{{harv|Rapp|1991|loc=§6}}; however, these are typically comparable | |||
in complexity to the method for the exact solution given above | |||
{{harv|Jekeli|2012|loc=§2.1.4}}. | |||
=== Behavior of geodesics === | |||
[[File:Closed geodesics on an ellipsoid of revolution.svg|thumb| | |||
Fig. 8. Meridians and the equator are the only closed | |||
geodesics. (For the very flattened ellipsoids, there are other closed | |||
geodesics; see Figs. 13 and 14).]] | |||
{{multiple image | |||
|align=right | |||
|direction=vertical | |||
|width=220 | |||
|image1=Latitude vs longitude for geodesic on an oblate ellipsoid.svg | |||
|caption1=Fig. 9. Latitude as a function of longitude for a single cycle of the geodesic from one northward equatorial crossing to the next. | |||
|image2=Long geodesic on an oblate ellipsoid.svg | |||
|caption2=Fig. 10. Following the geodesic on the ellipsoid for about 5 circuits. | |||
|image3=Really long geodesic on an oblate ellipsoid.svg | |||
|caption3=Fig. 11. The same geodesic after about 70 circuits. | |||
|header=Geodesic on an oblate ellipsoid (''f'' = 1/50) with α<sub>0</sub> = 45°. | |||
}} | |||
[[File:Long geodesic on a prolate ellipsoid.svg|thumb| | |||
Fig. 12. | |||
Geodesic on a prolate ellipsoid (''f'' = −1/50) with α<sub>0</sub> = 45°. Compare with | |||
Fig. 10.]] | |||
Before solving for the geodesics, it is worth reviewing their behavior. | |||
Fig. 8 shows the simple closed geodesics which consist of the | |||
meridians (green) and the equator (red). (Here the qualification | |||
"simple" means that the geodesic closes on itself without an intervening | |||
self-intersection.) This follows from the equations for the geodesics | |||
given in the previous section. | |||
For meridians, we have α<sub>0</sub> = 0 and Eq. (5) | |||
becomes λ = ω + λ<sub>0</sub>, i.e., the longitude will | |||
vary the same way as for a sphere, jumping by π each time | |||
the geodesic crosses the pole. The distance, Eq. (4), reduces to | |||
the length of an arc of an ellipse with semi-axes ''a'' and | |||
''b'' (as expected), expressed in terms of parametric latitude, | |||
β. | |||
The equator (β = 0</sub> on the auxiliary sphere, | |||
φ = 0 on the ellipsoid) corresponds to | |||
α<sub>0</sub> = ½π. The distance reduces to the arc of a | |||
circle of radius ''b'' (and ''not'' ''a''), | |||
''s'' = ''b''σ, while the longitude simplifies to | |||
λ = (1 − ''f'')σ + λ<sub>0</sub>. A geodesic that is nearly | |||
equatorial will intersect the equator at intervals of | |||
π''b''. As a consequence, the maximum length of a | |||
equatorial geodesic which is also a shortest path is π''b'' | |||
on an oblate ellipsoid (on a prolate ellipsoid, the maximum length is | |||
π''a''). | |||
All other geodesics are typified by Figs. 9 to 11. | |||
Figure 9 shows latitude as a function of longitude for a geodesic | |||
starting on the equator with α<sub>0</sub> = 45°. A full | |||
cycle of the geodesic, from one northward crossing of the equator to the | |||
next is shown. The equatorial crossings are called ''nodes'' and the | |||
points of maximum or minimum latitude are called ''vertices''; the | |||
vertex latitudes are given by | |||
|β| = ±(½π − |α<sub>0</sub>|). | |||
The latitude is an odd, resp. even, function of the longitude about the | |||
nodes, resp. vertices. The geodesic completes one full oscillation in | |||
latitude before the longitude has increased by 360°. | |||
Thus, on each successive northward crossing of the equator (see | |||
Fig. 10), λ falls short of a full circuit of | |||
the equator by approximately 2π ''f'' sinα<sub>0</sub> (for a | |||
prolate ellipsoid, this quantity is negative and λ | |||
completes more that a full circuit; see Fig. 12). For nearly all | |||
values of α<sub>0</sub>, the geodesic will fill that portion of | |||
the ellipsoid between the two vertex latitudes (see | |||
Fig. 11). | |||
{{multiple image | |||
|align=right | |||
|direction=vertical | |||
|width=220 | |||
|image1=Non-standard closed geodesics on an ellipsoid of revolution 1.svg | |||
|caption1=Fig. 13. Side view. | |||
|image2=Non-standard closed geodesics on an ellipsoid of revolution 2.svg | |||
|caption2=Fig. 14. Top view. | |||
|header=Two additional closed geodesics for the oblate ellipsoid, ''b''/''a'' = 2/7. | |||
}} | |||
If the ellipsoid is sufficiently oblate, i.e., | |||
''b''/''a'' < ½, another class of simple closed geodesics is | |||
possible {{harv|Klingenberg|1982|loc=§3.5.19}}. Two such geodesics | |||
are illustrated in Figs. 13 and 14. Here | |||
''b''/''a'' = 2/7 and the equatorial azimuth, | |||
α<sub>0</sub>, for the green (resp. blue) geodesic is chosen to | |||
be 53.175° (resp. 75.192°), so that the geodesic completes 2 | |||
(resp. 3) complete oscillations about the equator on one circuit of the | |||
ellipsoid. | |||
=== Evaluation of the integrals === | |||
Solving the geodesic problems entails evaluating the integrals for the | |||
distance, ''s'', and the longitude, λ, | |||
Eqs. (4) and (5). In geodetic applications where | |||
''f'' is small, the integrals are typically evaluated as a | |||
series; for this purpose the second form of the longitude integral is | |||
preferred (since it avoids the near singular behavior of the first form | |||
when geodesics pass close to a pole). In both integrals, the integrand | |||
is an even periodic function of period π. Furthermore the | |||
term dependent on σ is multiplied by a small quantity | |||
''k''<sup>2</sup> = ''O''(''f''). As a consequence, the integrals can both be | |||
written in the form | |||
:<math> | |||
I = B_0 \sigma + \sum_{j=1}^\infty B_j \sin 2j\sigma | |||
</math> | |||
where ''B''<sub>0</sub> = 1 + ''O''(''f'') and ''B''<sub>''j''</sub> = ''O''(''f'' <sup>''j''</sup>). Series | |||
expansions for ''B''<sub>''j''</sub> can readily be found and the result | |||
truncated so that only terms which are ''O''(''f'' <sup>''J''</sup>) and larger are | |||
retained.{{refn| Nowadays, the necessary algebraic manipulations, | |||
expanding in a Taylor series, integration, and performing trigonometric | |||
simplifications, can be carrying using a [[computer algebra system]]. | |||
Earlier, {{harvtxt|Levallois|Dupuy|1952}} gave recurrence relations for | |||
the series in terms of [[Wallis' integrals]] and | |||
{{harvtxt|Pittman|1986}} describes a similar method.}} | |||
(Because the longitude integral is multiplied by | |||
''f'', it is typically only necessary to retain terms up to | |||
''O''(''f'' <sup>''J''−1</sup>) in that integral.) This prescription is | |||
followed by many authors {{harv|Legendre|1806}} {{harv|Oriani|1806}} | |||
{{harv|Bessel|1825}} {{harv|Helmert|1880}} {{harv|Rainsford|1955}} | |||
{{harv|Rapp|1993}}. {{harvtxt|Vincenty|1975a}} uses ''J'' = 3 | |||
which provides an accuracy of about 0.1 mm for the [[WGS84]] | |||
ellipsoid. {{harvtxt|Karney|2013}} gives expansions carried out for | |||
''J'' = 6 which suffices to provide full [[double precision]] | |||
accuracy for |''f''| ≤ 1/50. Trigonometric | |||
series of this type can be conveniently summed using | |||
[[Clenshaw summation]]. | |||
In order to solve the direct geodesic problem, it is necessary to find | |||
σ given ''s''. Since the integrand in the | |||
distance integral is positive, this problem has a unique root, which | |||
may be found using [[Newton's method]], noting that the required | |||
derivative is just the integrand of the distance integral. | |||
{{harvtxt|Oriani|1833}} instead uses [[series reversion]] so that | |||
σ can be found without iteration; | |||
{{harvtxt|Helmert|1880}} gives a similar series.{{refn| | |||
{{harvtxt|Legendre|1806|loc=Art. 13}} also gives a series | |||
for σ in terms of ''s''; but this is | |||
not suitable for large distances.}} The reverted series | |||
converges somewhat slower that the direct series and, if | |||
|''f''| > 1/100, | |||
{{harvtxt|Karney|2013|loc=addenda}} supplements the reverted series with | |||
one step of Newton's method to maintain accuracy. | |||
{{harvtxt|Vincenty|1975a}} instead relies on a simpler (but slower) | |||
function iteration to solve for σ. | |||
It is also possible to evaluate the integrals (4) and (5) | |||
by numerical quadrature {{harv|Saito|1970}} {{harv|Saito|1979}} | |||
{{harv|Sjöberg|Shirazian|2012}} or to apply numerical techniques for the | |||
solution of the ordinary differential equations, Eqs. (2) | |||
{{harv|Kivioja|1971}} {{harv|Thomas|Featherstone|2005}} {{harv|Panou et | |||
al.|2013}}. Such techniques can be used for arbitrary flattening | |||
''f''. However, if ''f'' is small, e.g., | |||
|''f''| ≤ 1/50, they do not offer the speed | |||
and accuracy of the series expansions described above. Furthermore, for | |||
arbitrary ''f'', the evaluation of the integrals in terms of | |||
elliptic integrals (see below) also provides a fast and accurate | |||
solution. On the other hand, {{harvtxt|Mathar|2007}} has tackled the | |||
more complex problem of geodesics on the surface at a constant altitude, | |||
''h'', above the ellipsoid by solving the corresponding | |||
ordinary differential equations, Eqs. (2) with | |||
[ρ, ν] replaced by [ρ + ''h'', ν + ''h'']. | |||
[[File:Legendre.jpg|thumb|150px|[[Adrien-Marie Legendre|A. M. Legendre]]]] | |||
[[File:Arthur Cayley.jpg|thumb|150px|[[Arthur Cayley]]]] | |||
Geodesics on an ellipsoid was an early application of | |||
[[elliptic integrals]]. In particular, | |||
{{harvtxt|Legendre|1811|log=§§126–129}} writes the | |||
integrals, Eqs. (4) and (5), as | |||
:(6)<math>{\color{white}.}\qquad | |||
\displaystyle | |||
\frac sb = E(\sigma, ik), | |||
</math> | |||
:(7)<math>{\color{white}.}\qquad | |||
\begin{align} | |||
\lambda &= (1 - f) \sin\alpha_0 G(\sigma, \cos^2\alpha_0, ik) \\ | |||
&= \chi | |||
- \frac{e'^2}{\sqrt{1+e'^2}}\sin\alpha_0 H(\sigma, -e'^2, ik), \\ | |||
\end{align} | |||
</math> | |||
where | |||
:<math> | |||
\tan\chi = \sqrt{\frac{1+e'^2}{1+k^2\sin^2\sigma}}\tan\omega, | |||
</math> | |||
and | |||
:<math> | |||
\begin{align} | |||
G(\phi,\alpha^2,k) &= \int_0^\phi | |||
\frac{\sqrt{1 - k^2\sin^2\theta}}{1 - \alpha^2\sin^2\theta}\,d\theta\\ | |||
&=\frac{k^2}{\alpha^2}F(\phi, k) | |||
+\biggl(1-\frac{k^2}{\alpha^2}\biggr)\Pi(\phi, \alpha^2, k),\\ | |||
H(\phi, \alpha^2, k) | |||
&= \int_0^\phi | |||
\frac{\cos^2\theta}{(1-\alpha^2\sin^2\theta)\sqrt{1-k^2\sin^2\theta}} | |||
\,d\theta \\ | |||
&= | |||
\frac1{\alpha^2} F(\phi, k) + | |||
\biggl(1 - \frac1{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k), | |||
\end{align} | |||
</math> | |||
and ''F''(φ, ''k''), ''E''(φ, ''k''), and | |||
Π(φ, α<sup>2</sup>, ''k''), are | |||
[[Elliptic integral|incomplete elliptic integrals]] in the | |||
notation of | |||
{{harvtxt|DLMF|2010|loc=[http://dlmf.nist.gov/19.2.ii §19.2(ii)]}}.{{refn| | |||
1=Despite the presence of ''i'' = √−1, the elliptic | |||
integrals in Eqs. (6) and (7) are real.}}{{refn| | |||
{{harvtxt|Rollins|2010}} obtains different, but equivalent, expressions | |||
in terms of elliptic integrals.}} | |||
The first formula for the longitude in Eq. (7) follows directly | |||
from the first form of Eq. (5). The second formula in | |||
Eq. (7), due to {{harvtxt|Cayley|1870}}, is more | |||
convenient for calculation since the elliptic integral appears in a | |||
small term. The equivalence of the two forms follows from | |||
{{harvtxt|DLMF|2010|loc=[http://dlmf.nist.gov/19.7.E8 Eq. (19.7.8)]}}. | |||
Fast algorithms for computing elliptic integrals are given by | |||
{{harvtxt|Carlson|1995}} in terms of | |||
[[Carlson symmetric form|symmetric elliptic integrals]]. | |||
Equation (6) is conveniently inverted using | |||
[[Newton's method]]. The use of elliptic integrals provides a good | |||
method of solving the geodesic problem for | |||
|''f''| > 1/50.{{refn|1= | |||
It is also possible to express the integrals in terms of | |||
[[Jacobi elliptic functions]] {{harv|Jacobi|1855}} {{harv|Luther|1855}} | |||
{{harv|Forsyth|1896}} {{harv|Thomas|1970|loc=Appendix 1}}. | |||
{{harvtxt|Halphen|1888}} gives the solution for the complex quantities | |||
''R'' exp(±''i''λ) = ''X'' ± ''iY'' in terms of | |||
[[Weierstrass functions|Weierstrass sigma and zeta]] functions. This | |||
form is of interest because the separate periods of latitude and | |||
longitude of the geodesic are captured in a single | |||
[[doubly periodic function]]; see also | |||
{{harvtxt|Forsyth|1927|loc=§75.}}}} | |||
=== Solution of the direct problem === | |||
The basic strategy for solving the geodesic problems on the ellipsoid is | |||
to map the problem onto the auxiliary sphere by converting | |||
φ, λ, and ''s'' to | |||
β, ω and σ, solve | |||
the corresponding great-circle problem on the sphere and transfer the | |||
results back to the ellipsoid. | |||
In implementing this program, we will frequently need to solve the | |||
"elementary" spherical triangle problem for ''NEP'' in Fig. | |||
6 with ''P'' replaced by either ''A'' (subscript | |||
1) or ''B'' (subscript 2). For this purpose, we can apply | |||
[[Spherical trigonometry#Napier's rules for quadrantal triangles| | |||
Napier's rules for quadrantal triangles]] to the triangle ''NEP'' | |||
on the auxiliary sphere which give | |||
:<math> | |||
\begin{align} | |||
\sin\alpha_0 &= \sin\alpha \cos\beta = \tan\omega \cot\sigma,\\ | |||
\cos\sigma &= \cos\beta \cos\omega = \tan\alpha_0 \cot\alpha,\\ | |||
\cos\alpha &= \cos\omega \cos\alpha_0 = \cot\sigma \tan\beta,\\ | |||
\sin\beta &= \cos\alpha_0 \sin\sigma = \cot\alpha \tan\omega,\\ | |||
\sin\omega &= \sin\sigma \sin\alpha = \tan\beta \tan\alpha_0. | |||
\end{align} | |||
</math> | |||
We can also stipulate that cosβ ≥ 0 and | |||
cosα<sub>0</sub> ≥ 0.{{refn|name=atan| | |||
When solving for σ, α, or | |||
ω using a formula for its tangent, the quadrant should | |||
be determined from the signs of the numerator of the expression for the | |||
tangent, e.g., using the [[atan2]] function.}} | |||
Implementing this plan for the direct problem is straightforward. We | |||
are given φ<sub>1</sub>, α<sub>1</sub>, and | |||
''s''<sub>12</sub>. From φ<sub>1</sub> we obtain | |||
β<sub>1</sub> (using the formula for the parametric latitude). | |||
We now solve the triangle problem with ''P'' = ''A'' and | |||
β<sub>1</sub> and α<sub>1</sub> given to find | |||
α<sub>0</sub>, σ<sub>1</sub>, and | |||
ω<sub>1</sub>.{{refn|1= | |||
If β<sub>1</sub> = 0 and α<sub>1</sub> = ±½π, the | |||
equation for σ<sub>1</sub> is indeterminate and | |||
σ<sub>1</sub> = 0 may be used.}} Use | |||
the distance and longitude equations, Eqs. (4) and | |||
(5), together with the known value of λ<sub>1</sub>, to | |||
find ''s''<sub>1</sub> and λ<sub>0</sub>. Determine | |||
''s''<sub>2</sub> = ''s''<sub>1</sub> + ''s''<sub>12</sub> and invert the distance equation to find | |||
σ<sub>2</sub>. Solve the triangle problem with ''P'' = ''B'' | |||
and α<sub>0</sub> and σ<sub>2</sub> given to find | |||
β<sub>2</sub>, ω<sub>2</sub>, and α<sub>2</sub>. | |||
Convert β<sub>2</sub> to φ<sub>2</sub> and substitute | |||
σ<sub>2</sub> and ω<sub>2</sub> into the longitude | |||
equation to give λ<sub>2</sub>. | |||
The overall method follows the procedure for | |||
[[Great-circle navigation#Finding way-points|solving the direct | |||
problem on a sphere]]. It is essentially the program laid out by | |||
{{harvtxt|Bessel|1825}},{{refn| | |||
{{harvtxt|Bessel|1825}} treated the longitude integral approximately in | |||
order to reduce the number of parameters in the equation from two to one | |||
so that it could be tabulated conveniently.}} | |||
{{harvtxt|Helmert|1880|loc=§5.9}}, and most subsequent | |||
authors. | |||
=== Solution of the inverse problem === | |||
The ease with which the direct problem can be solved results from the | |||
fact that given φ<sub>1</sub> and α<sub>1</sub>, we can | |||
immediately find α<sub>0</sub>, the parameter in the distance | |||
and longitude integrals, Eqs. (4) and (5). In | |||
the case of the inverse problem, we are given λ<sub>12</sub>, | |||
but we cannot easily relate this to the equivalent spherical angle | |||
ω<sub>12</sub> because α<sub>0</sub> is unknown. | |||
Thus, the solution of the problem requires that α<sub>0</sub> be | |||
found iteratively. Before tackling this, it is worth understanding | |||
better the behavior of geodesics, this time, keeping the starting point | |||
fixed and varying the azimuth. | |||
{{multiple image | |||
|align=right | |||
|direction=vertical | |||
|width=220 | |||
|image1=Geodesics and geodesic circles on an oblate ellipsoid.svg | |||
|caption1=Fig. 15. Geodesics, geodesic circles, and the cut locus. | |||
|image2=Unrolled geodesics on an oblate ellipsoid.svg | |||
|caption2=Fig. 16. The geodesics shown on a [[plate carrée projection]]. | |||
|image3=Geodesic longitude variation for an ellipsoid.svg | |||
|caption3=Fig. 17. λ<sub>12</sub> as a function of α<sub>1</sub> for φ<sub>1</sub> = −30° and φ<sub>2</sub> = 20°. | |||
|header=Geodesics from a single point (''f'' = 1/10, φ<sub>1</sub> = −30°) | |||
}} | |||
Suppose point ''A'' in the inverse problem has | |||
φ<sub>1</sub> = −30° and λ<sub>1</sub> = 0°. Fig. | |||
15 shows geodesics (in blue) emanating | |||
''A'' with α<sub>1</sub> a multiple of | |||
15° up to the point at which they cease to be shortest | |||
paths. (The flattening has been increased to | |||
1/10 in order to accentuate the ellipsoidal effects.) | |||
Also shown (in green) are curves of constant ''s''<sub>12</sub>, | |||
which are the geodesic circles centered ''A''. | |||
{{harvtxt|Gauss|1828}} showed that, on any surface, geodesics and | |||
geodesic circle intersect at right angles. The red line is the | |||
[[cut locus]], the locus of points which have multiple (two in this | |||
case) shortest geodesics from ''A''. On a sphere, the cut | |||
locus is a point. On an oblate ellipsoid (shown here), it is a segment | |||
of the circle of latitude centered on the point [[antipodes|antipodal]] | |||
to ''A'', φ = −φ<sub>1</sub>. The longitudinal | |||
extent of cut locus is approximately | |||
λ<sub>12</sub> ∈ [π − ''f'' π cosφ<sub>1</sub>, π + ''f'' π cosφ<sub>1</sub>]. If | |||
''A'' lies on the equator, φ<sub>1</sub> = 0, this relation | |||
is exact and as a consequence the equator is only a shortest geodesic if | |||
|λ<sub>12</sub>| ≤ (1 − ''f'')π. For a prolate | |||
ellipsoid, the cut locus is a segment of the anti-meridian centered on | |||
the point antipodal to ''A'', λ<sub>12</sub> = π, | |||
and this means that | |||
meridional geodesics stop being shortest paths before the antipodal | |||
point is reached. | |||
The solution of the inverse problem involves determining, for a given | |||
point ''B'' with latitude φ<sub>2</sub> and longitude | |||
λ<sub>2</sub> which blue and green curves it lies on; this | |||
determines α<sub>1</sub> and ''s''<sub>12</sub> respectively. | |||
In Fig. 16, the ellipsoid has been "rolled out" onto a | |||
[[plate carrée projection]]. Suppose φ<sub>2</sub> = 20°, the | |||
green line in the figure. Then as α<sub>1</sub> is varied | |||
between 0° and 180°, the longitude | |||
at which the geodesic intersects φ = φ<sub>2</sub> varies between | |||
0° and 180° (see Fig. 17). | |||
This behavior holds provided that | |||
|φ<sub>2</sub>| ≤ |φ<sub>1</sub>| (otherwise the | |||
geodesic does not reach φ<sub>2</sub> for some values of | |||
α<sub>1</sub>). Thus, the inverse problem may be solved by | |||
determining the value α<sub>1</sub> which results in the given | |||
value of λ<sub>12</sub> when the geodesic intersects the circle | |||
φ = φ<sub>2</sub>. | |||
This suggests the following strategy for solving the inverse problem | |||
{{harv|Karney|2013}}. | |||
Assume that the points ''A'' and ''B'' satisfy | |||
:(8)<math>{\color{white}.}\qquad | |||
\phi_1 \le 0, \quad \left|\phi_2\right| \le \left|\phi_1\right|, | |||
\quad 0 \le \lambda_{12} \le \pi. </math> | |||
(There is no loss of generality in this assumption, since the symmetries | |||
of the problem can be used to generate any configuration of points from | |||
such configurations.) | |||
# First treat the "easy" cases, geodesics which lie on a meridian or the equator. Otherwise... | |||
# Guess a value of α<sub>1</sub>. | |||
# Solve the so-called ''hybrid geodesic problem'', given φ<sub>1</sub>, φ<sub>2</sub>, and α<sub>1</sub> find λ<sub>12</sub>, ''s''<sub>12</sub>, and α<sub>2</sub>, corresponding to the ''first'' intersection of the geodesic with the circle φ = φ<sub>2</sub>. | |||
# Compare the resulting λ<sub>12</sub> with the desired value and adjust α<sub>1</sub> until the two values agree. This completes the solution. | |||
Each of these steps requires some discussion. | |||
1. For an oblate ellipsoid, the shortest geodesic lies on a meridian if | |||
either point lies on a pole or if λ<sub>12</sub> = 0 or | |||
±π. The shortest geodesic follows the equator if | |||
φ<sub>1</sub> = φ<sub>2</sub> = 0 and | |||
|λ<sub>12</sub>| ≤ (1 − ''f'')π. For a prolate | |||
ellipsoid, the meridian is no longer the shortest geodesic if | |||
λ<sub>12</sub> = ±π and the points are close to antipodal | |||
(this will be discussed in the next section). There is no longitudinal | |||
restriction on equatorial geodesics. | |||
2. In most cases a suitable starting value of α<sub>1</sub> is | |||
found by solving | |||
[[Solution of triangles#Two sides and the included angle given| | |||
the spherical inverse problem]]{{refn|name=atan}} | |||
:<math>\tan\alpha_1 = \frac | |||
{\cos\beta_2\sin\omega_{12}} | |||
{\cos\beta_1\sin\beta_2 - \sin\beta_1\cos\beta_2 \cos\omega_{12}},</math> | |||
with ω<sub>12</sub> = λ<sub>12</sub>. This may be a bad | |||
approximation if ''A'' and ''B'' are nearly antipodal | |||
(both the numerator and denominator in the formula above become small); | |||
however, this may not matter (depending on how step 4 is handled). | |||
3. The solution of the hybrid geodesic problem is as follows. It starts | |||
the same way as the solution of the direct problem, solving the triangle | |||
''NEP'' with ''P'' = ''A'' to find α<sub>0</sub>, | |||
σ<sub>1</sub>, ω<sub>1</sub>, and | |||
λ<sub>0</sub>.{{refn|1=If φ<sub>1</sub> = φ<sub>2</sub> = 0, take | |||
sinσ<sub>1</sub> = sinω<sub>1</sub> = −0, consistent with the relations | |||
(8); this gives σ<sub>1</sub> = ω<sub>1</sub> = −π.}} Now find | |||
α<sub>2</sub> from | |||
sinα<sub>2</sub> = sinα<sub>0</sub>/cosβ<sub>2</sub>, taking | |||
cosα<sub>2</sub> ≥ 0 (corresponding to the first, northward, | |||
crossing of the circle φ = φ<sub>2</sub>). Next, | |||
σ<sub>2</sub> is given by | |||
tanσ<sub>2</sub> = tanβ<sub>2</sub>/cosα<sub>2</sub> and | |||
ω<sub>2</sub> by | |||
tanω<sub>2</sub> = tanσ<sub>2</sub>/sinα<sub>0</sub>.{{refn|name=atan}} | |||
Finally, use the distance and longitude equations, Eqs. (4) | |||
and (5), to find ''s''<sub>12</sub> and | |||
λ<sub>12</sub>.{{refn| | |||
The ordering in relations (8) automatically results in | |||
σ<sub>12</sub> > 0.}} | |||
4. In order to discuss how α<sub>1</sub> is updated, let us define | |||
the root-finding problem in more detail. The curve in Fig. 17 | |||
shows λ<sub>12</sub>(α<sub>1</sub>; φ<sub>1</sub>, φ<sub>2</sub>) where we regard | |||
φ<sub>1</sub> and φ<sub>2</sub> as parameters and | |||
α<sub>1</sub> as the independent variable. We seek the value of | |||
α<sub>1</sub> which is the root of | |||
:<math> g(\alpha_1) \equiv | |||
\lambda_{12}(\alpha_1;\phi_1,\phi_2) - \lambda_{12} = 0, | |||
</math> | |||
where ''g''(0) ≤ 0 and ''g''(π) ≥ 0. In fact, | |||
there is a unique root in the interval α<sub>1</sub> ∈ [0, π]. | |||
Any of a number of [[root-finding algorithms]] can be used to solve such | |||
an equation. {{harvtxt|Karney|2013}} uses [[Newton's method]], which | |||
requires a good starting guess; however it may be supplemented by a | |||
[[fail-safe]] method, such as the [[bisection method]], to guarantee | |||
convergence. | |||
[[File:F-R Helmert 1.jpg|thumb|150px|[[Friedrich Robert Helmert|F. R. Helmert]]]] | |||
An alternative method for solving the inverse problem is given by | |||
{{harvtxt|Helmert|1880|loc=§5.13}}. Let us rewrite the | |||
Eq. (5) as | |||
:<math>\begin{align} | |||
\lambda_{12} &= \omega_{12} | |||
- f\sin\alpha_0 | |||
\int_{\sigma_1}^{\sigma_2}\frac | |||
{2-f}{1 + (1-f)\sqrt{1 + k^2\sin^2\sigma'}} | |||
\,d\sigma'\\ | |||
&= \omega_{12} | |||
- f\sin\alpha_0 I(\sigma_1, \sigma_2; \alpha_0). | |||
\end{align} | |||
</math> | |||
Helmert's method entails assuming that | |||
ω<sub>12</sub> = λ<sub>12</sub>, solving the resulting problem on | |||
auxiliary sphere, and obtaining an updated estimate of | |||
ω<sub>12</sub> using | |||
:<math> | |||
\omega_{12} = \lambda_{12} + f\sin\alpha_0 I(\sigma_1, \sigma_2; \alpha_0). | |||
</math> | |||
This process is repeated until convergence. {{harvtxt|Vincenty|1975a}} | |||
uses this method in his solution of the inverse problem. The drawbacks | |||
of this method are that convergence is slower than obtained using | |||
Newton's method (as described above) and, more seriously, that the | |||
process fails to converge at all for nearly antipodal points. In a | |||
subsequent report, {{harvtxt|Vincenty|1975b}} attempts to cure this | |||
defect; but he is only partially successful. | |||
The shortest distance returned by the solution of the inverse problem is | |||
(obviously) uniquely defined. However, if ''B'' lies on the | |||
cut locus of ''A'' there are multiple azimuths which yield | |||
the same shortest distance. Here is a catalog of those cases: | |||
* φ<sub>1</sub> = −φ<sub>2</sub> (with neither point at a pole). If α<sub>1</sub> = α<sub>2</sub>, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by interchanging α<sub>1</sub> and α<sub>2</sub>. (This occurs when λ<sub>12</sub> ≈ ±π for oblate ellipsoids.) | |||
* λ<sub>12</sub> = ±π (with neither point at a pole). If α<sub>1</sub> = 0 or ±π, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by negating α<sub>1</sub> and α<sub>2</sub>. (This occurs when φ<sub>1</sub> + φ<sub>2</sub> ≈ 0 for prolate ellipsoids.) | |||
* ''A'' and ''B'' are at opposite poles. There are infinitely many geodesics which can be generated by varying the azimuths so as to keep α<sub>1</sub> + α<sub>2</sub> constant. (For spheres, this prescription applies when ''A'' and ''B'' are antipodal.) | |||
=== Differential behavior of geodesics === | |||
[[File:Carl Friedrich Gauss.jpg|thumb|150px|[[Carl Friedrich Gauss|C. F. Gauss]]]] | |||
[[File:Elwin Bruno Christoffel.JPG|thumb|150px|[[Elwin Christoffel|E. B. Christoffel]]]] | |||
Various problems involving geodesics require knowing their behavior | |||
when they are perturbed. This is useful in trigonometric adjustments | |||
{{harv|Ehlert|1993}}, | |||
determining the physical properties of signals which follow geodesics, | |||
etc. Consider a reference geodesic, parameterized by ''s'' the | |||
length from the northward equator crossing, and a second geodesic a small | |||
distance ''t''(''s'') away from it. {{harvtxt|Gauss|1828}} showed that | |||
''t''(''s'') obeys the | |||
[[Differential geometry of surfaces#Gauss–Jacobi equation| | |||
Gauss-Jacobi equation]] | |||
:(9)<math>{\color{white}.}\qquad | |||
\displaystyle\frac{d^2t(s)}{ds^2} = K(s) t(s), </math> | |||
[[File:Definition of reduced length and geodesic scale.svg|thumb| | |||
Fig. 18. | |||
Definition of reduced length and geodesic scale.]] | |||
where ''K''(''s'') is the [[Gaussian curvature]] at ''s''. | |||
The solution may be expressed as the sum of two independent solutions | |||
:<math> t(s_2) = C m(s_1,s_2) + D M(s_1,s_2) </math> | |||
where | |||
:<math> | |||
\begin{align} | |||
m(s_1, s_1) &= 0, \quad | |||
\left.\frac{dm(s_1,s_2)}{ds_2}\right|_{s_2 = s_1} = 1,\\ | |||
M(s_1, s_1) &= 1, \quad | |||
\left.\frac{dM(s_1,s_2)}{ds_2}\right|_{s_2 = s_1} = 0. | |||
\end{align} | |||
</math> | |||
We shall abbreviate ''m''(''s''<sub>1</sub>, ''s''<sub>2</sub>) = ''m''<sub>12</sub>, the so-called | |||
''reduced length'', and ''M''(''s''<sub>1</sub>, ''s''<sub>2</sub>) = ''M''<sub>12</sub>, the | |||
''geodesic scale''.{{refn| | |||
{{harvtxt|Bagratuni|1962|loc=§17}} uses the term "coefficient of | |||
convergence of ordinates" for the geodesic scale.}} | |||
Their basic definitions are illustrated in | |||
Fig. 18. {{harvtxt|Christoffel|1869}} made an extensive study of | |||
their properties. The reduced length obeys a reciprocity relation, | |||
:<math>m_{12} + m_{21} = 0.</math> | |||
Their derivatives are | |||
:<math> | |||
\begin{align} | |||
\frac{d m_{12}}{d s_2} &= M_{21},\\ | |||
\frac{d M_{12}}{d s_2} &= -\frac{1 - M_{12}M_{21}}{m_{12}}. | |||
\end{align} | |||
</math> | |||
Assuming that points 1, 2, and 3 lie on the same geodesic, then the | |||
following addition rules apply {{harv|Karney|2013}}, | |||
:<math> | |||
\begin{align} | |||
m_{13} &= m_{12} M_{23} + m_{23} M_{21},\\ | |||
M_{13} &= M_{12} M_{23} - (1 - M_{12} M_{21}) \frac{m_{23}}{m_{12}},\\ | |||
M_{31} &= M_{32} M_{21} - (1 - M_{23} M_{32}) \frac{m_{12}}{m_{23}}. | |||
\end{align} | |||
</math> | |||
The reduced length and the geodesic scale are components of the | |||
[[Jacobi field]]. | |||
The [[Ellipsoid of revolution#Curvature|Gaussian curvature for an ellipsoid of revolution]] | |||
is | |||
:<math> | |||
K = \frac{(1-e^2\sin^2\phi)^2}{b^2} | |||
= \frac{b^2}{a^4(1-e^2\cos^2\beta)^2}. | |||
</math> | |||
{{harvtxt|Helmert|1880|loc=Eq. (6.5.1.)}} solved the Gauss-Jacobi | |||
equation for this case obtaining | |||
:<math> | |||
\begin{align} | |||
m_{12}/b &= \sqrt{1 + k^2\sin^2\sigma_2}\, \cos\sigma_1 \sin\sigma_2 | |||
- \sqrt{1 + k^2\sin^2\sigma_1}\, \sin\sigma_1 \cos\sigma_2 \\ | |||
&\quad - \cos\sigma_1 \cos\sigma_2 \bigl(J(\sigma_2) - J(\sigma_1)\bigr),\\ | |||
M_{12} &= \cos\sigma_1 \cos\sigma_2 | |||
+ \frac{\sqrt{1 + k^2\sin^2\sigma_2}}{\sqrt{1 + k^2\sin^2\sigma_1}} | |||
\sin\sigma_1 \sin\sigma_2 \\ | |||
&\quad - \frac{\sin\sigma_1 \cos\sigma_2 | |||
\bigl(J(\sigma_2) - J(\sigma_1)\bigr)} | |||
{\sqrt{1 + k^2\sin^2\sigma_1}}, | |||
\end{align} | |||
</math> | |||
where | |||
:<math> | |||
\begin{align} | |||
J(\sigma) &= | |||
\int_0^\sigma \frac{k^2\sin^2\sigma'}{\sqrt{1 + k^2\sin^2\sigma'}}\,d\sigma'\\ | |||
&= E(\sigma, ik) - F(\sigma, ik). | |||
\end{align} | |||
</math> | |||
As we see from Fig. 18 (top sub-figure), the separation of two | |||
geodesics starting at the same point with azimuths differing by | |||
''d''α<sub>1</sub> is ''m''<sub>12</sub> ''d''α<sub>1</sub>. On a closed | |||
surface such as an ellipsoid, we expect ''m''<sub>12</sub> to oscillate | |||
about zero. Indeed, if the starting point of a geodesic is a pole, | |||
φ<sub>1</sub> = ½π, then the reduced length is the radius | |||
of the circle of latitude, | |||
''m''<sub>12</sub> = ''a'' cosβ<sub>2</sub> = ''a'' sinσ<sub>12</sub>. Similarly, for a | |||
meridional geodesic starting on the equator, | |||
φ<sub>1</sub> = α<sub>1</sub> = 0, we have | |||
''M''<sub>12</sub> = cosσ<sub>12</sub>. In the typical case, these | |||
quantities oscillate with a period of about 2π in | |||
σ<sub>12</sub> and grow linearly with distance at a rate | |||
proportional to ''f''. In trigonometric adjustments over small | |||
areas, it may be possible to approximate ''K''(''s'') in | |||
Eq. (9) by a constant ''K''. In this limit, the | |||
solutions for ''m''<sub>12</sub> and ''M''<sub>12</sub> are the same | |||
as for a sphere of radius 1/√''K'', namely, | |||
:<math>m_{12} = \sin(\sqrt K s_{12})/\sqrt K, \quad | |||
M_{12} = \cos(\sqrt K s_{12}).</math> | |||
To simplify the discussion of shortest paths in this paragraph we | |||
consider only geodesics with ''s''<sub>12</sub> > 0. The point at | |||
which ''m''<sub>12</sub> becomes zero is the point | |||
[[conjugate point|conjugate]] to the starting point. In order | |||
for a geodesic between ''A'' and ''B'', of length | |||
''s''<sub>12</sub>, to be a shortest path it must satisfy the | |||
Jacobi condition {{harv|Jacobi|1837}} | |||
{{harv|Forsyth|1927|loc=§§26–27}} | |||
{{harv|Bliss|1916}}, that there is | |||
no point conjugate to ''A'' between ''A'' and | |||
''B''. If this condition is not satisfied, then there is a | |||
''nearby'' path (not necessarily a geodesic) which is shorter. Thus, | |||
the Jacobi condition is a local property of the geodesic and is only a | |||
necessary condition for the geodesic being a global shortest path. | |||
Necessary and sufficient conditions for a geodesic being the shortest | |||
path are: | |||
* for an oblate ellipsoid, |σ<sub>12</sub>| ≤ π; | |||
* for a prolate ellipsoid, |λ<sub>12</sub>| ≤ π, if α<sub>0</sub> ≠ 0; if α<sub>0</sub> = 0, the supplemental condition ''m''<sub>12</sub> ≥ 0 is required if |λ<sub>12</sub>| = π. | |||
The latter condition above can be used to determine whether the shortest | |||
path is a meridian in the case of a prolate ellipsoid with | |||
|λ<sub>12</sub>| = π. The derivative required to | |||
solve the inverse method using Newton's method, | |||
∂λ<sub>12</sub>(α<sub>1</sub>; φ<sub>1</sub>, φ<sub>2</sub>) / ∂α<sub>1</sub>, | |||
is given in terms of the reduced length | |||
{{harv|Karney|2013|loc=Eq. (46)}}. | |||
=== Geodesic map projections === | |||
Two map projections are defined in terms of geodesics. They are based | |||
on polar and rectangular geodesic coordinates on the surface | |||
{{harv|Gauss|1828}}. The polar coordinate system | |||
(''r'', θ) is centered on some point ''A''. The | |||
coordinates of another point ''B'' are given by | |||
''r'' = ''s''<sub>12</sub> and θ = ½π − α<sub>1</sub> and | |||
these coordinates are used to find the projected coordinates on a plane | |||
map, ''x'' = ''r'' cosθ and ''y'' = ''r'' sinθ. The | |||
result is the familiar [[azimuthal equidistant projection]]; in the | |||
field of the [[differential geometry of surfaces]], it is called the | |||
[[exponential map]]. Due to the basic properties of geodesics | |||
{{harv|Gauss|1828}}, lines of constant ''r'' and lines of | |||
constant θ intersect at right angles on the | |||
surface. The scale of the projection in the radial direction is unity, | |||
while the scale in the azimuthal direction is | |||
''s''<sub>12</sub>/''m''<sub>12</sub>. | |||
The rectangular coordinate system (''x'', ''y'') uses a reference | |||
geodesic defined by ''A'' and α<sub>1</sub> as the | |||
''x'' axis. The point (''x'', ''y'') is found by traveling | |||
a distance ''s''<sub>13</sub> = ''x'' from ''A'' along the reference | |||
geodesic to an intermediate point ''C'' and then turning | |||
½π counter-clockwise and traveling along a | |||
geodesic a distance ''s''<sub>32</sub> = ''y''. If ''A'' is on the | |||
equator and α<sub>1</sub> = ½π, this gives the | |||
[[equidistant cylindrical projection]]. If α<sub>1</sub> = 0, | |||
this gives the [[Cassini projection|Cassini-Soldner projection]]. | |||
[[:fr:Carte de Cassini|Cassini's map of France]] placed ''A'' at | |||
the [[Paris Observatory]]. Due to the basic properties of geodesics | |||
{{harv|Gauss|1828}}, lines of constant ''x'' and lines of | |||
constant ''y'' intersect at right angles on the surface. The | |||
scale of the projection in the ''y'' direction is unity, while | |||
the scale in the ''x'' direction is 1/''M''<sub>32</sub>. | |||
The [[gnomonic projection]] is a projection of the sphere where all | |||
geodesics (i.e., great circles) map to straight lines (making it a | |||
convenient [[Great-circle navigation#Gnomonic chart|aid to navigation]]). | |||
Such a projection is only possible for surfaces of constant | |||
[[Gaussian curvature]] {{harv|Beltrami|1865}}. Thus a projection in | |||
which geodesics map to straight lines is not possible for an ellipsoid. | |||
However, it is possible to construct an ellipsoidal gnomonic projection | |||
in which this property ''approximately'' holds | |||
{{harv|Karney|2013|loc=§8}}. On the sphere, the gnomonic | |||
projection is the limit of a doubly azimuthal projection, a projection | |||
preserving the azimuths from two points ''A'' and | |||
''B'', as ''B'' approaches ''A''. Carrying | |||
out this limit in the case of a general surface yields an azimuthal | |||
projection in which the distance from the center of projection is given | |||
by ρ = ''m''<sub>12</sub>/''M''<sub>12</sub>. Even though geodesics are only | |||
approximately straight in this projection, all geodesics through the | |||
center of projection ''are'' straight. The projection can then be used to | |||
give an iterative but rapidly converging method of solving some problems | |||
involving geodesics, in particular, finding the intersection of two | |||
geodesics and finding the shortest path from a point to a geodesic. | |||
The [[Hammer retroazimuthal projection]] is a variation of the azimuthal | |||
equidistant projection {{harv|Hammer|1910}}. A geodesic is constructed | |||
from a central point ''A'' to some other point ''B''. | |||
The polar coordinates of the projection of ''B'' are | |||
''r'' = ''s''<sub>12</sub> and θ = ½π − α<sub>2</sub> | |||
(which depends on the azimuth at ''B'', instead of at | |||
''A''). This can be used to determine the direction from an | |||
arbitrary point to some fixed center. {{harvtxt|Hinks|1929}} suggested | |||
another application: if the central point ''A'' is a beacon, | |||
such as the [[Rugby Clock]], then at an unknown location ''B'' | |||
the range and the bearing to ''A'' can be measured and the | |||
projection can be used to estimate the location of ''B''. | |||
=== Envelope of geodesics === | |||
{{multiple image | |||
|align=right | |||
|direction=horizontal | |||
|width=220 | |||
|image1=Envelope of geodesics on an oblate ellipsoid.svg | |||
|caption1=Fig. 19. The envelope of geodesics from a point ''A'' at φ<sub>1</sub> = −30°. | |||
|image2=Four geodesics connecting two points on an oblate ellipsoid.svg | |||
|caption2=Fig. 20. The four geodesics connecting ''A'' and a point ''B'', φ<sub>2</sub> = 26°, λ<sub>12</sub> = 175°. | |||
|header=Geodesics from a single point (''f'' = 1/10, φ<sub>1</sub> = −30°) | |||
}} | |||
The geodesics from a particular point ''A'' if continued | |||
past the cut locus form an envelope illustrated in Fig. 19. | |||
Here the geodesics for which α<sub>1</sub> is a multiple of | |||
3° are shown in light blue. (The geodesics are only | |||
shown for their first passage close to the antipodal point, not for | |||
subsequent ones.) Some geodesic circles are shown in green; these form | |||
cusps on the envelope. The cut locus is shown in red. The envelope is | |||
the locus of points which are conjugate to ''A''; points on the | |||
envelope may be computed by finding the point at which | |||
''m''<sub>12</sub> = 0 on a geodesic (and Newton's method can be used to | |||
find this point). {{harvtxt|Jacobi|1891}} calls this star-like figure | |||
produced by the envelope an [[astroid]]. | |||
Outside the astroid two geodesics intersect at each point; thus there | |||
are two geodesics (with a length approximately half the | |||
circumference of the ellipsoid) between ''A'' and these points. | |||
This corresponds to the situation on the sphere where there are "short" | |||
and "long" routes on a great circle between two points. Inside the | |||
astroid four geodesics intersect at each point. Four such geodesics are | |||
shown in Fig. 20 where the geodesics are numbered in order of | |||
increasing length. (This figure uses the same position for | |||
''A'' as Fig. 15 and is drawn in the same projection.) | |||
The two shorter geodesics are ''stable'', i.e., ''m''<sub>12</sub> > 0, | |||
so that there is no nearby path connecting the two points which is | |||
shorter; the other two are unstable. Only the shortest line (the first | |||
one) has σ<sub>12</sub> ≤ π. All the geodesics are tangent | |||
to the envelope which is shown in green in the figure. A similar set of | |||
geodesics for the WGS84 ellipsoid is given in this table | |||
{{harv|Karney|2012|loc=Table 1}}: | |||
{| class="wikitable" style="text-align: right;" | |||
|+ Geodesics for φ<sub>1</sub> = −30°, φ<sub>2</sub> = 29.9°, λ<sub>12</sub> = 179.8° (WGS84) | |||
|- | |||
!No.!!α<sub>1</sub> (°)!!α<sub>2</sub> (°)!!''s''<sub>12</sub> (m)!!σ<sub>12</sub> (°)!!''m''<sub>12</sub> (m) | |||
|- | |||
|1||161.890524736|| 18.090737246||19989832.8276||179.894971388|| 57277.3769 | |||
|- | |||
|2|| 30.945226882||149.089121757||20010185.1895||180.116378785|| 24240.7062 | |||
|- | |||
|3|| 68.152072881||111.990398904||20011886.5543||180.267429871||−22649.2935 | |||
|- | |||
|4||−81.075605986||−99.282176388||20049364.2525||180.630976969||−68796.1679 | |||
|} | |||
The approximate shape of the astroid is given by | |||
:<math> x^{2/3} + y^{2/3} = 1</math> | |||
or, in parametric form, | |||
:<math> x = \cos^3\theta, \quad y = \sin^3\theta.</math> | |||
The astroid is also the envelope of the family of lines | |||
:<math> \frac x{\cos\gamma} + \frac y{\sin\gamma} = 1, </math> | |||
where γ is a parameter. (These are | |||
generated by the rod of the [[trammel of Archimedes]].) This aids | |||
in finding a good starting guess for α<sub>1</sub> for Newton's | |||
method for in inverse problem in the case of nearly antipodal points | |||
{{harv|Karney|2013|loc=§5}}. | |||
The astroid is the (exterior) [[evolute]] of the geodesic circles | |||
centered at ''A''. Likewise the geodesic circles are | |||
[[involute]]s of the astroid. | |||
=== Area of a geodesic polygon === | |||
A ''geodesic polygon'' is a polygon whose sides are geodesics. The area of | |||
such a polygon may be found by first computing the area between a | |||
geodesic segment and the equator, i.e., the area of the quadrilateral | |||
''AFHB'' in Fig. 1 {{harv|Danielsen|1989}}. Once this | |||
area is known, the area of a polygon may be computed by summing the | |||
contributions from all the edges of the polygon. | |||
Here we develop the formula for the area ''S''<sub>12</sub> of | |||
''AFHB'' following {{harvtxt|Sjöberg|2006}}. The area of any | |||
closed region of the ellipsoid is | |||
:<math> T = \int dT = \int \frac1K \cos\phi\,d\phi\,d\lambda, | |||
</math> | |||
where ''dT'' is an element of surface area and ''K'' | |||
is the [[Gaussian curvature]]. Now the | |||
[[Gauss-Bonnet theorem]] applied to a geodesic polygon states | |||
:<math> | |||
\Gamma = \int K \,dT = \int \cos\phi\,d\phi\,d\lambda, | |||
</math> | |||
where | |||
:<math> | |||
\Gamma = 2\pi - \sum_j \theta_j | |||
</math> | |||
is the geodesic excess and θ<sub>''j''</sub> is the exterior angle at | |||
vertex ''j''. Multiplying the equation for Γ | |||
by ''R''<sub>2</sub><sup>2</sup>, where ''R''<sub>2</sub> is the | |||
[[Earth radius#Authalic radius|authalic radius]], and subtracting this | |||
from the equation for ''T'' gives{{refn| | |||
{{harvtxt|Sjöberg|2006}} multiplies Γ by | |||
''b''<sup>2</sup> instead of ''R''<sub>2</sub><sup>2</sup>. However this leads to a | |||
singular integrand {{harv|Karney|2012|loc=§15}}.}} | |||
:<math> | |||
\begin{align} | |||
T &= | |||
R_2^2 \,\Gamma + \int \biggl(\frac1K - R_2^2\biggr)\cos\phi\,d\phi\,d\lambda\\ | |||
&=R_2^2 \,\Gamma + \int \biggl( | |||
\frac{b^2}{(1 - e^2\sin^2\phi)^2} - R_2^2 | |||
\biggr)\cos\phi\,d\phi\,d\lambda, | |||
\end{align} | |||
</math> | |||
where the [[Spheroid#Curvature|value of ''K'' for an ellipsoid]] | |||
has been substituted. | |||
Applying this formula to the quadrilateral ''AFHB'', noting | |||
that Γ = α<sub>2</sub> − α<sub>1</sub>, and performing | |||
the integral over φ gives | |||
:<math> | |||
\begin{align} | |||
S_{12}&=R_2^2 (\alpha_2-\alpha_1) | |||
+ b^2 \int_{\lambda_1}^{\lambda_2} \biggl( | |||
\frac1{2(1 - e^2\sin^2\phi)}\\ | |||
&\qquad\qquad{}+ | |||
\frac{\tanh^{-1}(e \sin\phi)}{2e \sin\phi} | |||
- \frac{R_2^2}{b^2}\biggr)\sin\phi | |||
\,d\lambda, | |||
\end{align} | |||
</math> | |||
where the integral is over the geodesic line (so that φ | |||
is implicitly a function of λ). Converting this into | |||
an integral over σ, we obtain | |||
:<math> | |||
\begin{align} | |||
S_{12} &= R_2^2 E_{12} - e^2a^2\cos\alpha_0 \sin\alpha_0 \times\\ | |||
&\quad \int_{\sigma_1}^{\sigma_2} | |||
\frac{t(e'^2) - t(k^2\sin^2\sigma)}{e'^2-k^2\sin^2\sigma} | |||
\frac{\sin\sigma}2 \,d\sigma, | |||
\end{align} | |||
</math> | |||
where | |||
:<math> | |||
t(x) = x + \sqrt{x^{-1} + 1}\,\sinh^{-1}\!\sqrt x, | |||
</math> | |||
and the notation ''E''<sub>12</sub> = α<sub>2</sub> − α<sub>1</sub> is used for | |||
the geodesic excess. | |||
The integral can be expressed as a series valid for small ''f'' | |||
{{harv|Danielsen|1989}} {{harv|Karney|2013|loc=§6 and addendum}}. | |||
The area of a geodesic polygon is given by summing ''S''<sub>12</sub> | |||
over its edges. This result holds provided that the polygon does not | |||
include a pole; if it does 2π ''R''<sub>2</sub><sup>2</sup> must be added to the | |||
sum. If the edges are specified by their vertices, then a | |||
[[Spherical excess|convenient expression]] | |||
for ''E''<sub>12</sub> is | |||
:<math> | |||
\tan\frac{E_{12}}2 = | |||
\frac{\sin\tfrac12 (\beta_2 + \beta_1)} | |||
{\cos\tfrac12 (\beta_2 - \beta_1)} \tan\frac{\omega_{12}}2. | |||
</math> | |||
This result follows from one of [[Napier's analogies]]. | |||
=== Software implementations === | |||
An implementation of Vincenty's algorithm in [[Fortran]] is provided by | |||
{{harvtxt|NGS|2012}}. Version 3.0 includes Vincenty's treatment of | |||
nearly antipodal points {{harv|Vincenty|1975b}}. | |||
[[Vincenty's formulae|Vincenty's original formulas]] are used in many | |||
geographic information systems. Except for nearly antipodal points | |||
(where the inverse method fails to converge), this method is accurate to | |||
about 0.5 mm for the WGS84 ellipsoid. | |||
The algorithms given in {{harvtxt|Karney|2013}} are included in | |||
GeographicLib {{harv|Karney|2013b}}. These are accurate to about | |||
15 nanometers for WGS84. Implementations in several | |||
languages ([[C++]], [[C (programming language)|C]], Fortran, | |||
[[Java (programming language)|Java]], [[JavaScript]], | |||
[[Python (programming language)|Python]], [[Matlab]], and | |||
[[Maxima (software)|Maxima]]) are included. In addition to solving the | |||
basic geodesic problem, this library can return ''m''<sub>12</sub>, | |||
''M''<sub>12</sub>, ''M''<sub>21</sub>, and ''S''<sub>12</sub>. A | |||
[[command-line utility]], <code>GeodSolve</code>, for geodesic | |||
calculations is included. As of version 4.9.0, the [[PROJ.4]] library | |||
for cartographic projections includes the C implementation. This is | |||
exposed in the two command-line utilities, <code>geod</code> and | |||
<code>invgeod</code>, and in the library itself. These algorithms have | |||
also been implemented in [[IDL (programming language)|IDL]] and | |||
[[C Sharp (programming language)|C#]]. | |||
The solution of the geodesic problems in terms of elliptic integrals is | |||
included in GeographicLib (in C++ only), e.g., via the <code>-E</code> | |||
option to <code>GeodSolve</code>. This method of solution is about | |||
2–3 time slower than using series expansions; however it | |||
provides accurate solutions for ellipsoids of revolution with | |||
''b''/''a'' ∈ [0.01, 100] {{harv|Karney|2013|loc=addenda}}. | |||
== Geodesics on a triaxial ellipsoid == | |||
Solving the geodesic problem for an ellipsoid of revolution is, from the | |||
mathematical point of view, relatively simple: because of symmetry, | |||
geodesics have a constant of the motion, given by Clairaut's relation | |||
allowing the problem to be reduced to | |||
[[Quadrature (mathematics)|quadrature]]. By the early 19th century | |||
(with the work of Legendre, [[Barnaba Oriani|Oriani]], Bessel, et al.), | |||
there was a complete understanding of the properties of geodesics on an | |||
ellipsoid of revolution. | |||
On the other hand, geodesics on a [[triaxial ellipsoid]] (with 3 unequal | |||
axes) have no obvious constant of the motion and thus represented a | |||
challenging "unsolved" problem in the first half of the 19th | |||
century. In a remarkable paper, {{harvtxt|Jacobi|1839}} discovered a | |||
constant of the motion allowing this problem to be reduced to quadrature | |||
also {{harv|Klingenberg|1982|loc=§3.5}}.{{refn| | |||
This section is adapted from the documentation for GeographicLib | |||
{{harv|Karney|2013b| | |||
loc=[http://geographiclib.sourceforge.net/1.32/triaxial.html geodesics on a triaxial ellipsoid]}}}}{{refn| | |||
Even though Jacobi and {{harvtxt|Weierstrass|1861}} | |||
use terrestrial geodesics as the motivation for their | |||
work, a triaxial ellipsoid approximates the earth only slightly better | |||
than an ellipsoid of revolution. A better approximation to the shape of | |||
the earth is given by the [[geoid]]. However geodesics on a surface of | |||
the complexity of the geoid are partly [[Chaos theory|chaotic]] | |||
{{harv|Waters|2011}}.}} | |||
=== Triaxial coordinate systems === | |||
[[File:Gaspard monge litho delpech.jpg|thumb|150px|[[Gaspard Monge]]]] | |||
[[File:Charles Dupin.jpeg|thumb|150px|[[Charles Dupin]]]] | |||
The key to the solution is expressing the problem in the "right" | |||
coordinate system. Consider the ellipsoid defined by | |||
:<math> | |||
h = \frac{X^2}{a^2} + \frac{Y^2}{b^2} + \frac{Z^2}{c^2} = 1, | |||
</math> | |||
where (''X'',''Y'',''Z'') are Cartesian coordinates centered on the | |||
ellipsoid and, without loss of generality, ''a'' ≥ ''b'' ≥ ''c'' > 0. A | |||
point on the surface is specified by a latitude and longitude. The | |||
''geographical'' latitude and longitude (φ, λ) are | |||
defined by | |||
:<math> | |||
\frac{\nabla h}{\left| \nabla h\right|} = \left( | |||
\begin{array}{c} \cos\phi \cos\lambda \\ \cos\phi \sin\lambda \\ \sin\phi | |||
\end{array}\right). | |||
</math> | |||
The ''parametric'' latitude and longitude (φ′, λ′) | |||
are defined by | |||
:<math> | |||
\begin{align} | |||
X &= a \cos\phi' \cos\lambda', \\ | |||
Y &= b \cos\phi' \sin\lambda', \\ | |||
Z &= c \sin\phi'. | |||
\end{align} | |||
</math> | |||
Jacobi employed the ''ellipsoidal'' latitude and longitude | |||
(β, ω) defined by | |||
[[File:Triaxial ellipsoid coordinate system.svg|thumb| | |||
Fig. 21. | |||
[[Ellipsoidal coordinates]].]] | |||
:<math> | |||
\begin{align} | |||
X &= a \cos\omega | |||
\frac{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}} | |||
{\sqrt{a^2 - c^2}}, \\ | |||
Y &= b \cos\beta \sin\omega, \\ | |||
Z &= c \sin\beta | |||
\frac{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}} | |||
{\sqrt{a^2 - c^2}}. | |||
\end{align} | |||
</math> | |||
In the limit ''b'' → ''a'', β | |||
becomes the parametric latitude for an oblate ellipsoid, so the use of | |||
the symbol β is consistent with the previous sections. | |||
However ω is ''different'' from the spherical | |||
longitude defined above.{{refn| | |||
The limit ''b'' → ''c'' gives a prolate ellipsoid with | |||
ω playing the role of the parametric latitude.}} | |||
Grid lines of constant β (in blue) and | |||
ω (in green) are given in Fig. 21. In contrast | |||
to (φ, λ) and (φ′, λ′), | |||
(β, ω) is an [[orthogonal]] coordinate system: the | |||
grid lines intersect at right angles. The principal sections of the | |||
ellipsoid, defined by ''X'' = 0 and ''Z'' = 0 are shown in | |||
red. The third principal section, ''Y'' = 0, is covered by the | |||
lines β = ±90° and ω = 0° or | |||
±180°. These lines meet at four | |||
[[umbilical point]]s (two of which are visible in this figure) where the | |||
[[principal curvature|principal radii of curvature]] are equal. Here | |||
and in the other figures in this section the parameters of the ellipsoid | |||
are ''a'':''b'':''c'' = 1.01:1:0.8, and it is viewed in an orthographic | |||
projection from a point above φ = 40°, | |||
λ = 30°. | |||
The grid lines of the ellipsoidal coordinates may be interpreted in three | |||
different ways | |||
# They are "lines of curvature" on the ellipsoid, i.e., they are parallel to the directions of principal curvature {{harv|Monge|1796}}. | |||
# They are also intersections of the ellipsoid with [[confocal ellipsoidal coordinates|confocal systems of hyperboloids of one and two sheets]] {{harv|Dupin|1813|loc = [http://books.google.com/books?id=j40AAAAAMAAJ&pg=PA297 Part 5]}}. | |||
# Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points {{harv|Hilbert|Cohn-Vossen|1952|p=188}}. For example, the lines of constant β in Fig. 21 can be generated with the familiar [[Gardener's ellipse|string construction for ellipses]] with the ends of the string pinned to the two umbilical points. | |||
Conversions between these three types of latitudes and longitudes | |||
and the Cartesian coordinates are simple algebraic exercises. | |||
The differential equations for a geodesic in ellipsoidal coordinates are | |||
:<math> | |||
\begin{align} | |||
\frac{d\beta}{ds} &= | |||
\frac1{\sqrt{(a^2-b^2)\sin^2\omega + (b^2-c^2)\cos^2\beta}} | |||
\frac | |||
{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}} | |||
{\sqrt{b^2 \sin^2\beta + c^2 \cos^2\beta}} \cos\alpha,\\ | |||
\frac{d\omega}{ds} &= | |||
\frac1{\sqrt{(a^2-b^2)\sin^2\omega + (b^2-c^2)\cos^2\beta}} | |||
\frac | |||
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}} | |||
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega}} \sin\alpha,\\ | |||
\frac{d\alpha}{ds} &= | |||
\frac1{((a^2-b^2)\sin^2\omega + (b^2-c^2)\cos^2\beta)^{3/2}}\times\\ | |||
&\quad\biggl(\frac | |||
{(a^2-b^2) \cos\omega\sin\omega | |||
\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}} | |||
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega}} \cos\alpha \\ | |||
&\qquad+\frac | |||
{(b^2-c^2) \cos\beta \sin\beta | |||
\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}} | |||
{\sqrt{b^2\sin^2\beta + c^2\cos^2\beta}} \sin\alpha\biggr). | |||
\end{align} | |||
</math> | |||
=== Jacobi's solution === | |||
[[File:Carl Jacobi.jpg|thumb|150px|[[C. G. J. Jacobi]]]] | |||
[[File:Joseph liouville.jpeg|thumb|150px|[[Joseph Liouville]]]] | |||
[[File:Darboux.jpg|thumb|150px|[[Jean-Gaston Darboux|J. G. Darboux]]]] | |||
Jacobi showed that the geodesic equations, expressed in ellipsoidal | |||
coordinates, are separable. Here is how he recounted his discovery to | |||
his friend and neighbor Bessel {{harv|Jacobi|1839|loc=Letter to Bessel}}, | |||
<blockquote> The day before yesterday, I reduced to quadrature the problem of geodesic lines on an ''ellipsoid with three unequal axes''. They are the simplest formulas in the world, [[Abelian integral]]s, which become the well known elliptic integrals if 2 axes are set equal.<br> | |||
[[Königsberg]], 28th Dec. '38. | |||
</blockquote> | |||
The solution given by {{harvtxt|Jacobi|1839}} is | |||
:<math> | |||
\begin{align} | |||
\delta &= \int \frac | |||
{\sqrt{b^2\sin^2\beta + c^2\cos^2\beta}\,d\beta} | |||
{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta} | |||
\sqrt{(b^2-c^2)\cos^2\beta - \gamma}}\\ | |||
&\quad - | |||
\int \frac | |||
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega}\,d\omega} | |||
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2} | |||
\sqrt{(a^2-b^2)\sin^2\omega + \gamma}}. | |||
\end{align} | |||
</math> | |||
As Jacobi notes "a function of the angle β equals | |||
a function of the angle ω. These two functions are | |||
just Abelian integrals..." Two constants δ and | |||
γ appear in the solution. Typically | |||
δ is zero if the lower limits of the integrals are | |||
taken to be the starting point of the geodesic and the direction of the | |||
geodesics is determined by γ. However for geodesics | |||
that start at an umbilical points, we have γ = 0 and | |||
δ determines the direction at the umbilical point. | |||
The constant γ may be expressed as | |||
:<math> | |||
\gamma = (b^2-c^2)\cos^2\beta\sin^2\alpha-(a^2-b^2)\sin^2\omega\cos^2\alpha, | |||
</math> | |||
where α is the angle the geodesic makes with lines of | |||
constant ω. In the limit ''b'' → ''a'', | |||
this reduces to sinα cosβ = const., the | |||
familiar Clairaut relation. A nice derivation of Jacobi's result is | |||
given by {{harvtxt|Darboux|1894|loc=§§583–584}} where he | |||
gives the solution found by {{harvtxt|Liouville|1846}} for general quadratic | |||
surfaces. In this formulation, the distance along the geodesic, | |||
''s'', is found using | |||
:<math> | |||
\begin{align} | |||
\frac{ds}{(a^2-b^2)\sin^2\omega + (b^2-c^2)\cos^2\beta} | |||
&= \frac | |||
{\sqrt{b^2\sin^2\beta + c^2\cos^2\beta}\,d\beta} | |||
{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta} | |||
\sqrt{(b^2-c^2)\cos^2\beta - \gamma}}\\ | |||
&= \frac | |||
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega}\,d\omega} | |||
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2} | |||
\sqrt{(a^2-b^2)\sin^2\omega + \gamma}}. | |||
\end{align} | |||
</math> | |||
An alternative expression for the distance is | |||
:<math> | |||
\begin{align} | |||
ds | |||
&= \frac | |||
{\sqrt{b^2\sin^2\beta + c^2\cos^2\beta} | |||
\sqrt{(b^2-c^2)\cos^2\beta - \gamma}\,d\beta} | |||
{\sqrt{a^2 - b^2\sin^2\beta - c^2\cos^2\beta}}\\ | |||
&\quad {}+ \frac | |||
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega} | |||
\sqrt{(a^2-b^2)\sin^2\omega + \gamma}\,d\omega} | |||
{\sqrt{a^2\sin^2\omega + b^2\cos^2\omega - c^2}}. | |||
\end{align} | |||
</math> | |||
=== Survey of triaxial geodesics === | |||
{{multiple image | |||
|align=right | |||
|direction=horizontal | |||
|width=220 | |||
|image1=Circumpolar geodesic on a triaxial ellipsoid case A.svg | |||
|image2=Circumpolar geodesic on a triaxial ellipsoid case B.svg | |||
|header=Circumpolar geodesics, ω<sub>1</sub> = 0°, α<sub>1</sub> = 90°. | |||
|caption1=Fig. 22. β<sub>1</sub> = 45.1°. | |||
|caption2=Fig. 23. β<sub>1</sub> = 87.48°. | |||
}} | |||
On a triaxial ellipsoid, there are only 3 simple closed geodesics, the | |||
three principal sections of the ellipsoid given by ''X'' = 0, | |||
''Y'' = 0, and ''Z'' = 0.{{refn| | |||
If ''c''/''a'' < ½, there are other simple closed geodesics | |||
similar to those shown in Figs. 13 and 14 | |||
{{harv|Klingenberg|1982|loc=§3.5.19}}.}} | |||
To survey the other geodesics, it is convenient to consider geodesics | |||
which intersect the middle principal section, ''Y'' = 0, at right | |||
angles. Such geodesics are shown in Figs. 22–26, | |||
which use the same ellipsoid parameters and the same viewing direction | |||
as Fig. 21. In addition, the three principal ellipses are shown | |||
in red in each of these figures. | |||
If the starting point is β<sub>1</sub> ∈ (−90°, 90°), | |||
ω<sub>1</sub> = 0, and α<sub>1</sub> = 90°, then the | |||
geodesic encircles the ellipsoid in a "circumpolar" sense. The geodesic | |||
oscillates north and south of the equator; on each oscillation it | |||
completes slightly less that a full circuit around the ellipsoid | |||
resulting, in the typical case, in the geodesic filling the area bounded | |||
by the two latitude lines β = ±β<sub>1</sub>. Two examples | |||
are given in Figs. 22 and 23. Figure 22 shows | |||
practically the same behavior as for an oblate ellipsoid of revolution | |||
(because ''a'' ≈ ''b''); compare to Fig. 11. | |||
However, if the starting point is at a higher latitude (Fig. 22) | |||
the distortions resulting from ''a'' ≠ ''b'' are evident. All | |||
tangents to a circumpolar geodesic touch the confocal single-sheeted | |||
hyperboloid which intersects the ellipsoid at β = β<sub>1</sub> | |||
{{harv|Chasles|1846}} | |||
{{harv|Hilbert|Cohn-Vossen|1952|pp=223–224}}. | |||
{{multiple image | |||
|align=right | |||
|direction=horizontal | |||
|width=220 | |||
|image1=Transpolar geodesic on a triaxial ellipsoid case A.svg | |||
|image2=Transpolar geodesic on a triaxial ellipsoid case B.svg | |||
|header=Transpolar geodesics, β<sub>1</sub> = 90°, α<sub>1</sub> = 0°. | |||
|caption1=Fig. 24. ω<sub>1</sub> = 39.9°. | |||
|caption2=Fig. 25. ω<sub>1</sub> = 9.966°. | |||
}} | |||
If the starting point is β<sub>1</sub> = 90°, | |||
ω<sub>1</sub> ∈ (0°, 180°), and | |||
α<sub>1</sub> = 0°, then the geodesic encircles the ellipsoid | |||
in a "transpolar" sense. The geodesic oscillates east and west of the | |||
ellipse ''X'' = 0; on each oscillation it completes slightly more | |||
that a full circuit around the ellipsoid resulting, in the typical case, | |||
in the geodesic filling the area bounded by the two longitude lines | |||
ω = ω<sub>1</sub> and ω = 180° − ω<sub>1</sub>. | |||
If ''a'' = ''b'', all meridians are geodesics; the effect of | |||
''a'' ≠ ''b'' causes such geodesics to oscillate east and west. | |||
Two examples are given in Figs. 24 and 25. The constriction | |||
of the geodesic near the pole disappears in the limit | |||
''b'' → ''c''; in this case, the ellipsoid becomes a | |||
prolate ellipsoid and Fig. 24 would resemble Fig. 12 | |||
(rotated on its side). All tangents to a transpolar geodesic touch the | |||
confocal double-sheeted hyperboloid which intersects the ellipsoid at | |||
ω = ω<sub>1</sub>. | |||
[[File:Unstable umbilical geodesic on a triaxial ellipsoid.svg|thumb| | |||
Fig. 26. An umbilical geodesic, β<sub>1</sub> = 90°, | |||
ω<sub>1</sub> = 0°, α<sub>1</sub> = 45°.]] | |||
If the starting point is β<sub>1</sub> = 90°, | |||
ω<sub>1</sub> = 0° (an umbilical point), and | |||
α<sub>1</sub> = 45° (the geodesic leaves the ellipse | |||
''Y'' = 0 at right angles), then the geodesic repeatedly | |||
intersects the opposite umbilical point and returns to its starting | |||
point. However on each circuit the angle at which it intersects | |||
''Y'' = 0 becomes closer to 0° or | |||
180° so that asymptotically the geodesic lies on the | |||
ellipse ''Y'' = 0 {{harv|Hart|1849}} {{harv|Arnold|1989|p=265}}. | |||
This is shown in Fig. 26. Note that a single geodesic does not | |||
fill an area on the ellipsoid. All tangents to umbilical geodesics | |||
touch the confocal hyperbola which intersects the ellipsoid at the | |||
umbilic points. | |||
Umbilical geodesic enjoy several interesting properties. | |||
* Through any point on the ellipsoid, there are two umbilical geodesics. | |||
* The geodesic distance between opposite umbilical points is the same regardless of the initial direction of the geodesic. | |||
* Whereas the closed geodesics on the ellipses ''X'' = 0 and ''Z'' = 0 are stable (an geodesic initially close to and nearly parallel to the ellipse remains close to the ellipse), the closed geodesic on the ellipse ''Y'' = 0, which goes through all 4 umbilical points, is ''exponentially unstable''. If it is perturbed, it will swing out of the plane ''Y'' = 0 and flip around before returning to close to the plane. (This behavior may repeat depending on the nature of the initial perturbation.) | |||
If the starting point ''A'' of a geodesic is not an umbilical | |||
point, then its envelope is an astroid with two cusps lying on | |||
β = −β<sub>1</sub> and the other two on | |||
ω = ω<sub>1</sub> + π {{harv|Sinclair|2003}}. The cut locus | |||
for ''A'' is the portion | |||
of the line β = −β<sub>1</sub> between the cusps | |||
{{harv|Itoh|Kiyohara|2004}}. | |||
{{harv|Panou|2013}} gives a method for solving the inverse problem for a | |||
triaxial ellipsoid by directly integrating the system of | |||
ordinary differential equations for a geodesic. (Thus, it does not | |||
utilize Jacobi's solution.) | |||
== Applications == | |||
[[File:Karl Weierstrass.jpg|thumb|150px|[[Karl Weierstrass]]]] | |||
[[File:Henri Poincaré-2.jpg|thumb|150px|[[Henri Poincaré]]]] | |||
The direct and inverse geodesic problems no longer play the central role | |||
in geodesy that they once did. Instead of solving adjustment problems | |||
as a two-dimensional problem in spheroidal trigonometry, these problem | |||
are now solved by three-dimensional methods | |||
{{harv|Vincenty|Bowring|1978}}. Nevertheless, terrestrial geodesics | |||
still play an important role in several areas: | |||
* for measuring distances and areas in [[geographic information systems]]; | |||
* the definition of [[maritime boundaries]] {{harv|UNCLOS|2006}}; | |||
* in the rules of the [[Federal Aviation Administration]] for area navigation {{harv|RNAV|2007}}; | |||
* the method of measuring distances in the [[Fédération Aéronautique Internationale|FAI]] Sporting Code {{harv|FAI|2013}}. | |||
By the [[principle of least action]], many problems in physics can be | |||
formulated as a variational problem similar to that for geodesics. Indeed | |||
the geodesic problem is equivalent to the motion of a particle | |||
constrained to move on the surface, but otherwise subject to no forces | |||
{{harv|Laplace|1799a}} {{harv|Hilbert|Cohn-Vossen|1952|p=222}}. | |||
For this reason, | |||
geodesics on simple surfaces such as ellipsoids of revolution or | |||
triaxial ellipsoids are frequently used as "test cases" for exploring new | |||
methods. Examples include: | |||
* the development of elliptic integrals {{harv|Legendre|1811}} and [[elliptic functions]] {{harv|Weierstrass|1861}}; | |||
* the development of differential geometry {{harv|Gauss|1828}} {{harv|Christoffel|1869}}; | |||
* methods for solving systems of differential equations by a change of independent variables {{harv|Jacobi|1839}}; | |||
* the study of [[caustic (optics)|caustics]] {{harv|Jacobi|1891}}; | |||
* investigations into the number and stability of periodic orbits {{harv|Poincaré|1905}}; | |||
* in the limit ''c'' → 0, geodesics on a triaxial ellipsoid reduce to a case of [[dynamical billiards]]; | |||
* extensions to an arbitrary number of dimensions {{harv|Knörrer|1980}}; | |||
* geodesic flow on a surface {{harv|Berger|2010|loc=Chap. 12}}. | |||
== See also == | |||
* [[Geographical distance]] | |||
* [[Great-circle navigation]] | |||
* [[Geodesics]] | |||
* [[Geodesy]] | |||
* [[Rhumb line]] | |||
* [[Vincenty's formulae]] | |||
== Notes == | |||
{{reflist|30em}} | |||
== References == | |||
{{refbegin|30em}} | |||
*{{cite book | |||
|ref = harv |year = 1989 | |||
|last = Arnold |first = V. I. |authorlink = Vladimir Arnold | |||
|title = Mathematical Methods of Classical Mechanics | |||
|edition = 2nd | |||
|publisher = Springer-Verlag | |||
|others = Translated by K. Vogtmann & A. Weinstein | |||
|oclc = 4037141 | |||
|isbn = 978-0-387-96890-2 | |||
}} | |||
*{{cite book | |||
|ref = {{harvid|Bagratuni|1962}} |year = 1967 |origyear = 1962 | |||
|last = Bagratuni |first = G. V. | |||
|title = Course in Spheroidal Geodesy | |||
|url = http://geographiclib.sf.net/geodesic-papers/bagratuni67.pdf | |||
|postscript = . Translation of ''Курс сфероидической геодезии'' by U.S. Air Force ([http://www.dtic.mil/docs/citations/AD0650520 FTD-MT-64-390]) | |||
|oclc = 6150611 | |||
}} | |||
*{{cite doi | |||
|10.1007/BF03198517 | |||
|comment = Beltrami 1865 | |||
|noedit | |||
}} | |||
*{{cite book | |||
|ref = harv |year = 2010 | |||
|last = Berger |first = M. |authorlink = Marcel Berger | |||
|title = Geometry Revealed | |||
|publisher = Springer | |||
|others = Translated by L. J. Senechal | |||
|isbn = 978-3-540-70996-1 | |||
|doi = 10.1007/978-3-540-70997-8 | |||
}} | |||
*{{cite journal | |||
|ref = {{harvid|Bessel|1825}} |year = 2010 |origyear = 1825 | |||
|last = Bessel |first = F. W. |authorlink = Friedrich Bessel | |||
|doi = 10.1002/asna.201011352 | |||
|title = The calculation of longitude and latitude from geodesic measurements | |||
|journal = Astronomische Nachrichten | |||
|volume = 331 |issue = 8 |pages = 852–861 | |||
|arxiv = 0908.1824 | |||
|others = . Translated by C. F. F. Karney & R. E. Deakin | |||
|postscript = . English translation of [http://adsabs.harvard.edu/abs/1825AN......4..241B ''Astron. Nachr.'' '''4''', 241–254 (1825)]. [http://geographiclib.sourceforge.net/bessel-errata.html Errata]. | |||
}} | |||
*{{cite doi | |||
|10.1002/asna.18270051202 | |||
|comment = Bessel 1827 | |||
|noedit | |||
}} | |||
*{{cite doi | |||
|10.1090/S0002-9947-1916-1501037-4 | |||
|comment = Bliss 1916 | |||
|noedit | |||
}} | |||
*{{cite book | |||
|ref = harv |year = 1952 | |||
|last = Bomford |first = G. | |||
|title = Geodesy | |||
|publisher = Clarendon | |||
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}} | |||
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|title = The effect of the Earth's oblate spheroid shape on the accuracy of a time-of-arrival lightning ground strike locating system | |||
|url = http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19910023392_1991023392.pdf | |||
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}} | |||
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|ref = harv |year = 1735 | |||
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|title = De la carte de la France et de la perpendiculaire a la méridienne de Paris | |||
|trans_title = The map of France and the perpendicular to the meridian of Paris | |||
|language = French | |||
|journal = Mémoires de l'Académie Royale des Sciences de Paris 1733 | |||
|pages = 389–405 | |||
|url = http://books.google.com/books?id=GOAEAAAAQAAJ&pg=PA389 | |||
}} | |||
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|ref = harv |year = 1870 | |||
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|title = On the geodesic lines on an oblate spheroid | |||
|journal = Philosophical Magazine (4th ser.) | |||
|volume = 40 | |||
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|doi = 10.1080/14786447008640411 | |||
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}} | |||
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|ref = harv |year = 1846 | |||
|last = Chasles |first = M. |authorlink = Michel Chasles | |||
|title = Sur les lignes géodésiques et les lignes de courbure des surfaces du second degré | |||
|language = French | |||
|trans_title = Geodesic lines and the lines of curvature of the | |||
surfaces of the second degree | |||
|journal = Journal de Mathématiques Pures et Appliquées | |||
|volume = 11 | |||
|pages = 5–20 | |||
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}} | |||
*{{cite journal | |||
|ref = harv |year = 1869 | |||
|last = Christoffel |first = E. B. |authorlink = Elwin Bruno Christoffel | |||
|title = Allgemeine Theorie der geodätischen Dreiecke | |||
|trans_title = General theory of geodesic triangles | |||
|language = German | |||
|journal = Abhandlungen Königlichen Akademie der Wissenschaft zu Berlin | |||
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}} | |||
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|trans_title = Geometrical determination of the perpendicular to the meridian drawn by Jacques Cassini | |||
|language = French | |||
|journal = Mémoires de l'Académie Royale des Sciences de Paris 1733 | |||
|pages = 406–416 | |||
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}} | |||
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}} | |||
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|title = Leçons sur la théorie générale des surfaces | |||
|volume = 3 | |||
|language = French | |||
|trans_title = Lessons on the general theory of surfaces | |||
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}} | |||
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|title = NIST Handbook of Mathematical Functions | |||
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}} | |||
*{{cite book | |||
|ref = harv |year = 1813 | |||
|last = Dupin |first = P. C. F. |authorlink = Charles Dupin | |||
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|trans_title = Developments in geometry | |||
|publisher = Courcier | |||
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|trans_title = Methods for ellipsoidal triangulatioin | |||
|language = German | |||
|institution = [[:de:Deutsche Geodätische Kommission|Deutsche Geodätische Kommission]] | |||
|series = Reihe B: Angewandte Geodäsie, Heft Nr. 292 | |||
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}} | |||
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|trans_title = Elements of spheroidal trigonometry taken from the method of maxima and minima | |||
|language = French | |||
|journal = Mémoires de l'Académie Royale des Sciences de Berlin 1753 | |||
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}} | |||
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|ref = harv |year = 2013 | |||
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}} | |||
*{{cite journal | |||
|ref = harv |year = 1896 | |||
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|journal = Messenger of Mathematics | |||
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}} | |||
*{{cite book | |||
|ref = harv |year = 1927 | |||
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}} | |||
*{{cite book | |||
|ref = {{harvid|Gauss|1828}} |year = 1902 |origyear = 1828 | |||
|last = Gauss |first = C. F. |authorlink = Carl Friedrich Gauss | |||
|title = General Investigations of Curved Surfaces of 1827 and 1825 | |||
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|postscript = . [http://geographiclib.sourceforge.net/geodesic-papers/gauss28-en.pdf PDF]. English translation of [http://books.google.com/books?id=bX0AAAAAMAAJ&pg=PA3 ''Disquisitiones generales circa superficies curvas''] (Dieterich, Göttingen, 1828). | |||
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}} | |||
*{{cite book | |||
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|last = Halphen |first = G. H. |authorlink = Georges Henri Halphen | |||
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|trans_title = A Treatise on Elliptic Functions and their Applications | |||
|volume = 2 | |||
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}} | |||
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|ref = harv |year = 1910 | |||
|last = Hammer |first = E. |authorlink = :de:Ernst Hammer (Geodät) | |||
|title = Gegenazimutale Projektion | |||
|language = German | |||
|trans_title = Retroazimuthal projection | |||
|journal = Petermanns Geographische Mitteilungen | |||
|volume = 56 | |||
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}} | |||
*{{cite book | |||
|ref = harv |year = 1865 | |||
|last = Hansen |first = P. A. |authorlink = Peter Andreas Hansen | |||
|title = Geodätische Untersuchungen | |||
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|trans_title = Geodetic investigations | |||
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}} | |||
*{{cite journal | |||
|ref = harv |year = 1849 | |||
|last = Hart |first = A. S. |authorlink = Andrew Searle Hart | |||
|title = Geometrical demonstration of some properties of geodesic lines | |||
|journal = Cambridge and Dublin Mathematical Journal | |||
|volume = 4 | |||
|pages = 80–84 | |||
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}} | |||
*{{cite book | |||
|ref = {{harvid|Helmert|1880}} |year = 1964 |origyear = 1880 | |||
|last = Helmert |first = F. R. |authorlink = Friedrich Robert Helmert | |||
|title = Mathematical and Physical Theories of Higher Geodesy | |||
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|postscript = . English translation of [http://books.google.com/books?id=qt2CAAAAIAAJ ''Die Mathematischen und Physikalischen Theorieen der Höheren Geodäsie''], Vol. 1 (Teubner, Leipzig, 1880). | |||
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}} | |||
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|title = Geometry and the Imagination | |||
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|title = A Course of Mathematics in Three Volumes Composed for the Use of the Royal Military Academy | |||
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|work = Jacobi's Gesammelte Werke | |||
|title = Über die Curve, welche alle von einem Punkte ausgehenden geodätischen Linien eines Rotationsellipsoides berührt | |||
|publisher = Reimer | |||
|location = Berlin | |||
|trans_title = The envelope of geodesic lines emanating from | |||
a single point on an ellipsoid | |||
|language = German | |||
|pages = 72–87 | |||
|editor = K. T. W. Weierstrass (editor) |editor-link = Karl Weierstrass | |||
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|postscript = . Op. post., completed by [[Friedrich Heinrich Albert Wangerin|F. H. A. Wangerin]]. [http://geographiclib.sourceforge.net/geodesic-papers/jacobi-V7.pdf PDF]. | |||
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*{{citation | |||
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}} | |||
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*{{cite book | |||
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|others = Translated by [[Nathaniel Bowditch|N. Bowditch]] | |||
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}} | |||
*{{cite journal | |||
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|title = Analyse des triangles tracées sur la surface d'un sphéroïde | |||
|trans_title = Analysis of spheroidal triangles | |||
|language = French | |||
|journal = Mémoires de l'Institut National de France | |||
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}} | |||
*{{cite book | |||
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|last = Legendre |first = A. M. |authorlink = Adrien-Marie Legendre | |||
|title = Exercices de Calcul Intégral sur Divers Ordres de Transcendantes et sur les Quadratures | |||
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|publisher = Courcier | |||
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|title = Note sur le Calcul des Grandes Géodésiques | |||
|institution = Institut Géographique National | |||
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}} | |||
*{{cite journal | |||
|ref = harv |year = 1846 | |||
|last = Liouville |first = J. |authorlink = Joseph Liouville | |||
|title = Sur quelques cas particuliers où les équations du mouvement d'un point matériel peuvent s'intégrer | |||
|language = French | |||
|trans_title = Special cases where the equations of motion are integrable | |||
|journal = Journal de Mathématiques Pures et Appliquées | |||
|volume = 11 | |||
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|ref = harv |year = 1929 | |||
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|last2 = Schnirelmann |first2 = L. |authorlink2 = Lev Schnirelmann | |||
|title = Sur le problème de trois géodésiques fermées sur les surfaces de genre 0 | |||
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|trans_title = The problem of three closed geodesics on surfaces of genus 0 | |||
|journal = Comptes Rendus de l'Académie des Sciences de Paris | |||
|volume = 189 | |||
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}} | |||
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|10.1002/asna.18550422201 | |||
|comment = Luther 1855 | |||
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}} | |||
*{{cite arxiv | |||
|ref = harv |year= 2007 | |||
|last = Mathar |first = R. J. | |||
|title = Geodetic line at constant altitude above the ellipsoid | |||
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}} | |||
*{{cite encyclopedia | |||
|ref = {{harvid|Monge|1796}} |year = 1850 |origyear = 1796 | |||
|last = Monge |first = G. |authorlink = Gaspard Monge | |||
|title = Sur les lignes de courbure de la surface de l'ellipsoïde | |||
|language = French | |||
|trans_title = On the lines of curvature on the surface of the ellipsoid | |||
|work = Application de l'Analyse à la Géometrie | |||
|pages = 139–160 | |||
|url = http://books.google.com/books?id=Nf5zhlffjd0C&pg=PA139 | |||
|editor = J. Liouville (editor) |editor-link = Joseph Liouville | |||
|edition = 5th | |||
|publisher = Bachelier | |||
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}} | |||
*{{cite doi | |||
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|comment = Munk+Forbes 1989 | |||
|noedit | |||
}} | |||
*{{cite web | |||
|ref = {{harvid|NGS|2012}} |year = 2012 | |||
|title = Geodesic Utilities: Inverse and Forward | |||
|version = 3.0 | |||
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}} | |||
*{{cite book | |||
|ref = {{harvid|Newton|1687}} |year = 1848 |origyear = 1687 | |||
|last = Newton |first = I. |authorlink = Isaac Newton | |||
|title = The Mathematical Principles of Natural Philosophy | |||
|publisher = Adee | |||
|location = New York | |||
|others = Translated by A. Motte | |||
|postscript = . Book 3, Proposition 19, Problem 3, pp. 405–409. | |||
|url = http://books.google.com/books?id=KaAIAAAAIAAJ&pg=PA405 | |||
}} | |||
*{{cite journal | |||
|ref = harv |year = 1806 | |||
|last = Oriani |first = B. |authorlink = Barnaba Oriani | |||
|title = Elementi di trigonometria sferoidica, Pt. 1 | |||
|language = Italian | |||
|trans_title = Elements of spheroidal trigonometry | |||
|journal = Memorie dell'Istituto Nazionale Italiano | |||
|volume = 1 | |||
|number = 1 | |||
|pages = 118–198 | |||
|url = http://books.google.com/books?id=SydFAAAAcAAJ&pg=PA118 | |||
}} | |||
*{{cite journal | |||
|ref = harv |year = 1808 | |||
|last = Oriani |first = B. |authorlink = Barnaba Oriani | |||
|title = Elementi di trigonometria sferoidica, Pt. 2 | |||
|language = Italian | |||
|trans_title = Elements of spheroidal trigonometry | |||
|journal = Memorie dell'Istituto Nazionale Italiano | |||
|volume = 2 | |||
|number = 1 | |||
|pages = 1–58 | |||
|url = http://www.archive.org/stream/memoriedellistit21isti#page/1 | |||
}} | |||
*{{cite journal | |||
|ref = harv |year = 1810 | |||
|last = Oriani |first = B. |authorlink = Barnaba Oriani | |||
|title = Elementi di trigonometria sferoidica, Pt. 3 | |||
|language = Italian | |||
|trans_title = Elements of spheroidal trigonometry | |||
|journal = Memorie dell'Istituto Nazionale Italiano | |||
|volume = 2 | |||
|number = 2 | |||
|pages = 1–58 | |||
|url = http://www.archive.org/stream/memoriedellistit22isti#page/1 | |||
}} | |||
*{{cite doi | |||
|10.1002/asna.18260043201 | |||
|comment = Oriani 1826 | |||
|noedit | |||
}} | |||
*{{cite journal | |||
|ref = harv |year = 1833 | |||
|last = Oriani |first = B. |authorlink = Barnaba Oriani | |||
|title = Nota aggiunta agli elementi della trigonometria sferoidica | |||
|language = Italian | |||
|trans_title = Note added to the elements of spheroidal trigonometry | |||
|journal = Memorie dell'Imperiale Regio Istituto del Regno Lombardo-Veneto | |||
|volume = 4 | |||
|pages = 325–331 | |||
|url = http://books.google.com/books?id=6bsAAAAAYAAJ&pg=PA325 | |||
}} | |||
*{{cite doi | |||
|10.2478/jogs-2013-0007 | |||
|comment = Panou et al. 2013 | |||
|noedit | |||
}} | |||
*{{cite doi | |||
|10.2478/jogs-2013-0028 | |||
|comment = Panou 2013 | |||
|noedit | |||
}} | |||
*{{cite journal | |||
|ref = harv |year = 1986 | |||
|last = Pittman |first = M. E. | |||
|title = Precision direct and inverse solutions of the geodesic | |||
|journal = Surveying and Mapping | |||
|volume = 46 | |||
|issue = 1 | |||
|pages = 47–54 | |||
}} | |||
*{{cite doi | |||
|10.2307/1986219 | |||
|comment = Poincaré 1905 | |||
|noedit | |||
}} | |||
*{{cite doi | |||
|10.1007/BF02527187 | |||
|comment = Rainsford 1955 | |||
|noedit | |||
}} | |||
*{{citation | |||
|ref = harv |year = 1991 | |||
|last = Rapp |first = R. H. | |||
|title = Geometric geodesy, part I | |||
|publisher = Ohio State Univ. | |||
|url = http://hdl.handle.net/1811/24333 | |||
}} | |||
*{{citation | |||
|ref = harv |year = 1993 | |||
|last = Rapp |first = R. H. | |||
|title = Geometric geodesy, part II | |||
|publisher = Ohio State Univ. | |||
|url = http://hdl.handle.net/1811/24409 | |||
}} | |||
*{{cite techreport | |||
|ref = harv |year = 2007 | |||
|last = RNAV |authorlink = RNAV | |||
|title = Order 8260.54A, The United States Standard for Area Navigation | |||
|institution = U.S. Federal Aviation Administration | |||
|location = Washington, DC | |||
|url = http://www.faa.gov/documentLibrary/media/Order/8260.54A.pdf | |||
|postscript = . Appendix 2. | |||
}} | |||
*{{cite doi | |||
|10.1179/003962609X451663 | |||
|comment = Rollins 2010 | |||
|noedit | |||
}} | |||
*{{cite doi | |||
|10.1007/BF02522166 | |||
|comment = Saito 1970 | |||
|noedit | |||
}} | |||
*{{cite doi | |||
|10.1007/BF02521087 | |||
|comment = Saito 1979 | |||
|noedit | |||
}} | |||
*{{cite doi | |||
|10.1080/10586458.2003.10504515 | |||
|comment = Sinclair 2003 | |||
|noedit | |||
}} | |||
*{{cite doi | |||
|10.1179/003962606780732100 | |||
|comment = Sjöberg 2006 | |||
|noedit | |||
}} | |||
*{{cite doi | |||
|10.1061/(ASCE)SU.1943-5428.0000061 | |||
|comment = Sjöberg+Shirazian 2012 | |||
|noedit | |||
}} | |||
*{{cite doi | |||
|10.1061/(ASCE)0733-9453(2005)131:1(20) | |||
|comment = Thomas+Featherstone 2005 | |||
|noedit | |||
}} | |||
*{{cite techreport | |||
|ref = harv |year = 1970 | |||
|last = Thomas |first = P. D. | |||
|title = Spheroidal Geodesics, Reference Systems, & Local Geometry | |||
|url = http://www.dtic.mil/docs/citations/AD0703541 | |||
|institution = U.S. Naval Oceanographic Office | |||
|number = SP-138 | |||
}} | |||
*{{cite techreport | |||
|ref = harv |year = 2006 | |||
|last = UNCLOS |authorlink = UNCLOS | |||
|title = A Manual on Technical Aspects of the United Nations Convention on the Law of the Sea, 1982 | |||
|institution = International Hydrographic Bureau | |||
|edition = 4th | |||
|location = Monaco | |||
|url = http://www.iho.int/iho_pubs/CB/C-51_Ed4-EN.pdf | |||
}} | |||
*{{cite journal | |||
|ref = harv |year = 1975a | |||
|last = Vincenty |first = T. |authorlink = Thaddeus Vincenty | |||
|title = Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations | |||
|journal = Survey Review | |||
|volume = 23 | |||
|number = 176 | |||
|pages = 88–93 | |||
|url = http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf | |||
|postscript = . Addendum: Survey Review '''23''' (180): 294 (1976). | |||
}} | |||
*{{cite techreport | |||
|ref = harv |year = 1975b | |||
|last = Vincenty |first = T. |authorlink = Thaddeus Vincenty | |||
|title = Geodetic inverse solution between antipodal points | |||
|institution = DMAAC Geodetic Survey Squadron | |||
|url = http://geographiclib.sourceforge.net/geodesic-papers/vincenty75b.pdf | |||
|accessdate = 2011-07-28 | |||
}} | |||
*{{cite techreport | |||
|ref = harv |year = 1978 | |||
|last1 = Vincenty |first1 = T. |authorlink1 = Thaddeus Vincenty | |||
|last2 = Bowring |first2 = B. R. | |||
|title = Application of three-dimensional geodesy to adjustments of horizontal networks | |||
|institution = NOAA | |||
|number = NOS NGS-13 | |||
|url = http://www.ngs.noaa.gov/PUBS_LIB/ApplicationOfThreeDimensionalGeodesyToAdjustmentsOfHorizontalNetworks_TM_NOS_NGS13.pdf | |||
}} | |||
*{{cite doi | |||
|10.1016/j.physd.2011.11.010 | |||
|comment = Waters 2012 | |||
|noedit | |||
}} | |||
*{{cite journal | |||
|ref = harv |year = 1861 | |||
|last = Weierstrass |first = K. T. W. |authorlink = Karl Weierstrass | |||
|title = Über die geodätischen Linien auf dem dreiaxigen Ellipsoid | |||
|trans_title = Geodesic lines on a triaxial ellipsoid | |||
|language = German | |||
|journal = Monatsbericht Königlichen Akademie der Wissenschaft zu Berlin | |||
|pages = 986–997 | |||
|url = http://books.google.com/books?id=9O4GAAAAYAAJ&pg=PA257 | |||
|postscript = . [http://geographiclib.sourceforge.net/geodesic-papers/weierstrass-V1.pdf PDF]. | |||
}} | |||
{{refend}} | |||
== External links == | |||
* [http://geographiclib.sourceforge.net/geodesic-papers/biblio.html Online geodesic bibliography], approximately 150 books and articles on geodesics on ellipsoids together with links to online copies. | |||
* Implementations of {{harvtxt|Vincenty|1975a}} for oblate ellipsoids: | |||
** [http://www.ngs.noaa.gov/PC_PROD/Inv_Fwd/ NGS implementation], includes modifications described in {{harvtxt|Vincenty|1975b}}. | |||
** [http://www.ngs.noaa.gov/cgi-bin/Inv_Fwd/forward2.prl NGS online solution of the direct problem]. | |||
** [http://www.ga.gov.au/earth-monitoring/geodesy/geodetic-techniques/distance-calculation-algorithms.html Online calculator from Geoscience Australia]. | |||
** Javascript implementations of solutions to [http://www.movable-type.co.uk/scripts/latlong-vincenty-direct.html direct problem] and [http://www.movable-type.co.uk/scripts/latlong-vincenty.html inverse problem]. | |||
* Implementation of {{harvtxt|Karney|2013}} for ellipsoids of revolution in Geographiclib {{harv|Karney|2013b}}: | |||
** [http://geographiclib.sourceforge.net/ GeographicLib web site] for downloading library and documentation. | |||
** [http://geographiclib.sourceforge.net/html/GeodSolve.1.html GeodSolve(1)], [[man page]] for a utility for geodesic calculations. | |||
** [http://geographiclib.sourceforge.net/cgi-bin/GeodSolve An online version of GeodSolve]. | |||
** [http://geographiclib.sourceforge.net/html/Planimeter.1.html Planimeter(1)], man page for a utility for calculating the area of geodesic polygons. | |||
** [http://geographiclib.sourceforge.net/cgi-bin/Planimeter An online version of Planimeter]. | |||
** [http://geographiclib.sourceforge.net/scripts/geod-calc.html Javascript utility for direct and inverse problems and area calculations]. | |||
** [http://geographiclib.sourceforge.net/scripts/geod-google.html Drawing geodesics on Google Maps]. | |||
** [http://www.mathworks.com/matlabcentral/fileexchange/39108 Matlab implementation of the geodesic routines] (used for the figures for geodesics on ellipsoids of revolution in this article). | |||
* Geodesics on a triaxial ellipsoid: | |||
** [http://geographiclib.sourceforge.net/html/triaxial.html Additional notes about Jacobi's solution]. | |||
** [http://www.math.harvard.edu/~knill/caustic/exhibits/ellipsoid/index.html Caustics on an ellipsoid]. | |||
[[Category:Geodesy]] | |||
[[Category:Geodesic (mathematics)]] | |||
[[Category:Differential geometry]] | |||
[[Category:Calculus of variations]] | |||
Revision as of 20:20, 10 May 2013
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, i.e., the analogue of a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry Template:Harv.


If the earth is treated as a sphere, the geodesics are great circles (all of which are closed) and the problems reduce to ones in spherical trigonometry. However, Template:Harvtxt showed that the effect of the rotation of the earth results in its resembling a slightly oblate ellipsoid and, in this case, the equator and the meridians are the only closed geodesics. Furthermore, the shortest path between two points on the equator does not necessarily run along the equator. Finally, if the ellipsoid is further perturbed to become a triaxial ellipsoid (with three distinct semi-axes), then only three geodesics are closed and one of these is unstable.
The problems in geodesy are usually reduced to two main cases: the direct problem, given a starting point and an initial heading, find the position after traveling a certain distance along the geodesic; and the inverse problem, given two points on the ellipsoid find the connecting geodesic and hence the shortest distance between them. Because the flattening of the earth is small, the geodesic distance between two points on the earth is well approximated by the great-circle distance using the mean earth radius—the relative error is less than 1%. However, the course of the geodesic can differ dramatically from that of the great circle. As an extreme example, consider two points on the equator with a longitude difference of 179°59′; while the connecting great circle follows the equator, the shortest geodesics pass within 180 km of either pole (the flattening makes two symmetric paths passing close to the poles shorter than the route along the equator).
Aside from their use in geodesy and related fields such as navigation, terrestrial geodesics arise in the study of the propagation of signals which are confined (approximately) to the surface of the earth, for example, sound waves in the ocean Template:Harv and the radio signals from lightning Template:Harv. Geodesics are used to define some maritime boundaries, which in turn determine the allocation of valuable resources as such oil and mineral rights. Ellipsoidal geodesics also arise in other applications; for example, the propagation of radio waves along the fuselage of an aircraft, which can be roughly modeled as a prolate (elongated) ellipsoid Template:Harv.
Geodesics are an important intrinsic characteristic of curved surfaces. The sequence of progressively more complex surfaces, the sphere, an ellipsoid of revolution, and a triaxial ellipsoid, provide a useful family of surfaces for investigating the general theory of surfaces. Indeed Gauss's work on the survey of Hanover, which involved geodesics on an oblate ellipsoid, was a key motivation for his study of surfaces Template:Harv. Similarly the existence of three closed geodesics on a triaxial ellipsoid turns out to be a general property of closed, simply connected surfaces; this was conjectured by Template:Harvtxt and proved by Template:Harvtxt Template:Harv.
Geodesics on an ellipsoid of revolution
There are several ways of defining geodesics Template:Harv. A simple definition is as the shortest path between two points on a surface. However it is frequently more useful to define them as paths with zero geodesic curvature—i.e., the analogue of straight lines on a curved surface. This definition encompasses geodesics traveling so far across the ellipsoid's surface (somewhat less than half the circumference) that other distinct routes require less distance. Locally, these geodesics are still identical to the shortest distance between two points.
By the end of the 18th century, an ellipsoid of revolution (the term spheroid is also used) was a well-accepted approximation to the figure of the Earth. The adjustment of triangulation networks entailed reducing all the measurements to a reference ellipsoid and solving the resulting two-dimensional problem as an exercise in spheroidal trigonometry Template:Harv.
It is possible to reduce the various geodesic problems into one of two types. Consider two points: A at latitude φ1 and longitude λ1 and B at latitude φ2 and longitude λ2 (see Fig. 1). The connecting geodesic (from A to B) is AB, of length s12, which has azimuths α1 and α2 at the two endpoints.Template:Refn The two geodesic problems usually considered are:
- the direct geodesic problem or first geodesic problem, given A, α1, and s12, determine B and α2;
- the inverse geodesic problem or second geodesic problem, given A and B, determine s12, α1, and α2.
As can be seen from Fig. 1, these problems involve solving the triangle NAB given one angle, α1 for the direct problem and λ12 = λ2 − λ1 for the inverse problem, and its two adjacent sides. In the course of the 18th century these problems were elevated (especially in literature in the German language) to the principal geodesic problems Template:Harv.
For a sphere the solutions to these problems are simple exercises in spherical trigonometry, whose solution is given by formulas for solving a spherical triangle. (See the article on great-circle navigation.)


For an ellipsoid of revolution, the characteristic constant defining the geodesic was found by Template:Harvtxt. A systematic solution for the paths of geodesics was given by Template:Harvtxt and Template:Harvtxt (and subsequent papers in [[#Template:Harvid|1808]] and [[#Template:Harvid|1810]]). The full solution for the direct problem (complete with computational tables and a worked out example) is given by Template:Harvtxt.Template:Refn
Much of the early work on these problems was carried out by mathematicians—for example, Legendre, Bessel, and Gauss—who were also heavily involved in the practical aspects of surveying. Beginning in about 1830, the disciplines diverged: those with an interest in geodesy concentrated on the practical aspects such as approximations suitable for field work, while mathematicians pursued the solution of geodesics on a triaxial ellipsoid, the analysis of the stability of closed geodesics, etc.
During the 18th century geodesics were typically referred to as "shortest lines".Template:Refn The term "geodesic line" was coined by Template:Harvtxt:
Nous désignerons cette ligne sous le nom de ligne géodésique [We will call this line the geodesic line].
This terminology was introduced into English either as "geodesic line" or as "geodetic line", for example Template:Harv,
A line traced in the manner we have now been describing, or deduced from trigonometrical measures, by the means we have indicated, is called a geodetic or geodesic line: it has the property of being the shortest which can be drawn between its two extremities on the surface of the earth; and it is therefore the proper itinerary measure of the distance between those two points.
In its adoption by other fields "geodesic line", frequently shortened, to "geodesic", was preferred.Template:Refn
This section treats the problem on an ellipsoid of revolution (both oblate and prolate). The problem on a triaxial ellipsoid is covered in the next section.
Equations for a geodesic

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Here the equations for a geodesic are developed; these allow the geodesics of any length to be computed accurately. The following derivation closely follows that of Template:Harvtxt. Template:Harvtxt, Template:Harvtxt, and Template:Harvtxt also provide derivations of these equations.
Consider an ellipsoid of revolution with equatorial radius a and polar semi-axis b. Define the flattening f = (a − b)/a, eccentricity e2 = f(2 − f) second eccentricity e′ = e/(1 − f). (In most applications in geodesy, the ellipsoid is taken to be oblate, a > b; however, the theory applies without change to prolate ellipsoids, a < b, in which case f, e2, and e′2 are negative.)
Let an elementary segment of a path on the ellipsoid have length ds. From Figs. 2 and 3, we see that if its azimuth is α, then ds can is related to dφ and dλ by
where ρ is the meridional radius of curvature, R = ν cosφ is the radius of the circle of latitude φ, and ν is the normal radius of curvature. The elementary segment can therefore be expressed as
where φ′ = dφ/dλ and L depends on φ through ρ(φ) and R(φ). The length of an arbitrary path between (φ1, λ1) and (φ2, λ2) is given by
where φ is a function of λ satisfying φ(λ1) = φ1 and φ(λ2) = φ2. The shortest path or geodesic entails finding that function φ(λ) which minimizes s12. This is an exercise in the calculus of variations and the minimizing condition is given by the Beltrami identity,
Substituting for L and using Eq. (1) gives
Template:Harvtxt first found this relation, using a geometrical construction.Template:Refn Differentiating this relation and manipulating the result gives Template:Harv
This, together with Eqs. (1), leads to a system of ordinary differential equations for a geodesic
We can express R in terms of the parametric latitude, β, using
(see Fig. 4 for the geometrical construction), and Clairaut's relation then becomes
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for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed
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6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!
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This is the sine rule of spherical
trigonometry relating two sides of the triangle NAB (see
Fig. 5), NA = ½π − β1, and
NB = ½π − β2 and their opposite angles
B = π − α2 and A = α1.
In order to find the relation for the third side AB = σ12, the spherical arc length, and included angle N = ω12, the spherical longitude, it is useful to consider the triangle NEP representing a geodesic starting at the equator; see Fig. 6. In this figure, the variables referred to the auxiliary sphere are shown with the corresponding quantities for the ellipsoid shown in parentheses. Quantities without subscripts refer to the arbitrary point P; E, the point at which the geodesic crosses the equator in the northward direction, is used as the origin for σ, s and ω.
If the side EP is extended by moving P infinitesimally (see Fig. 7), we obtain
Combining Eqs. (1) and (3) gives differential equations for s and λ
Up to this point, we have not made use of the specific equations for an ellipsoid, and indeed the derivation applies to an arbitrary surface of revolution.Template:Refn Bessel now specializes to an ellipsoid in which R and Z are related by
where Z is the height above the equator (see Fig. 4). Differentiating this and setting dR/dZ = −sinφ/cosφ gives
eliminating Z from these equations, we obtain
This relation between β and φ can be written as
which is the normal definition of the parametric latitude on an ellipsoid. Furthermore, we have
so that the differential equations for the geodesic become
The last step is to use σ as the independent parameterTemplate:Refn in both of these differential equations and thereby to express s and λ as integrals. Applying the sine rule to the vertices E and G in the spherical triangle EGP in Fig. 6 gives
where α0 is the azimuth at E. Substituting this into the equation for ds/dσ and integrating the result gives
where
and the limits on the integral are chosen so that s(σ = 0) = 0. Template:Harvtxt pointed out that the equation for s is the same as the equation for the arc on an ellipse with semi-axes a(1 − e2 sin2α0)1/2 and b. In order to express the equation for λ in terms of σ, we write
which follows from Eq. (2) and Clairaut's relation. This yields
and the limits on the integrals are chosen so that λ = λ0 at the equator crossing, σ = 0.
In using these integral relations, we allow σ to increase continuously (not restricting it to a range [−π, π], for example) as the great circle, resp. geodesic, encircles the auxiliary sphere, resp. ellipsoid. The quantities ω, λ, and s are likewise allowed to increase without limit. Once the problem is solved, λ can be reduced to the conventional range.
This completes the solution of the path of a geodesic using the auxiliary sphere. By this device a great circle can be mapped exactly to a geodesic on an ellipsoid of revolution. However, because the equations for s and λ in terms of the spherical quantities depend on α0, the mapping is not a consistent mapping of the surface of the sphere to the ellipsoid or vice versa; instead, it should be viewed merely as a convenient tool for solving for a particular geodesic.
There are also several ways of approximating geodesics on an ellipsoid which usually apply for sufficiently short lines Template:Harv; however, these are typically comparable in complexity to the method for the exact solution given above Template:Harv.
Behavior of geodesics
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for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed
After obtaining a Sale License (topic to authorities circumstances meant to guard individuals shopping for property in Singapore), he might www.rmmonline.com proceed to sell units in his growth. Funding in the property market of Singapore is likely one of the few investment choices the place utilizing the bank's cash could not be any easier. The power of expats, to make a down payment, leverage the capital and consequently improve total return on investment, is excessive in Singapore. PRs who own a HDB flat must sell their flat inside six months of buying a non-public residential property in Singapore. EVERLASTING residents (PRs) now face unprecedented limits on their skill to purchase property in Singapore. New Condo in 2013 March Foreigners are eligible for Singapore greenback mortgage loan. housing grant
Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013
6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!
BuyRentSingaporeProperty.com is dedicated in providing the newest residential launches, Developer's sales, new launches, resale, HIGH tasks and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are sometimes in the vicinity of the city fringe, in areas corresponding to Newton and Novena Tanjong Rhu. These new launch projects are popular due to their proximity to town heart, the place the enterprise and entertainment centers and all the amenities they provide, but they're usually cheaper than the City core region tasks. Why submit cheques to register for new property launches in Singapore? Riversails Apartment @ Higher Serangoon LakeVille @ Jurong Lakeside Condo by MCL Ferra Condo @ Leonie Hill
Before solving for the geodesics, it is worth reviewing their behavior. Fig. 8 shows the simple closed geodesics which consist of the meridians (green) and the equator (red). (Here the qualification "simple" means that the geodesic closes on itself without an intervening self-intersection.) This follows from the equations for the geodesics given in the previous section.
For meridians, we have α0 = 0 and Eq. (5) becomes λ = ω + λ0, i.e., the longitude will vary the same way as for a sphere, jumping by π each time the geodesic crosses the pole. The distance, Eq. (4), reduces to the length of an arc of an ellipse with semi-axes a and b (as expected), expressed in terms of parametric latitude, β.
The equator (β = 0 on the auxiliary sphere, φ = 0 on the ellipsoid) corresponds to α0 = ½π. The distance reduces to the arc of a circle of radius b (and not a), s = bσ, while the longitude simplifies to λ = (1 − f)σ + λ0. A geodesic that is nearly equatorial will intersect the equator at intervals of πb. As a consequence, the maximum length of a equatorial geodesic which is also a shortest path is πb on an oblate ellipsoid (on a prolate ellipsoid, the maximum length is πa).
All other geodesics are typified by Figs. 9 to 11. Figure 9 shows latitude as a function of longitude for a geodesic starting on the equator with α0 = 45°. A full cycle of the geodesic, from one northward crossing of the equator to the next is shown. The equatorial crossings are called nodes and the points of maximum or minimum latitude are called vertices; the vertex latitudes are given by |β| = ±(½π − |α0|). The latitude is an odd, resp. even, function of the longitude about the nodes, resp. vertices. The geodesic completes one full oscillation in latitude before the longitude has increased by 360°. Thus, on each successive northward crossing of the equator (see Fig. 10), λ falls short of a full circuit of the equator by approximately 2π f sinα0 (for a prolate ellipsoid, this quantity is negative and λ completes more that a full circuit; see Fig. 12). For nearly all values of α0, the geodesic will fill that portion of the ellipsoid between the two vertex latitudes (see Fig. 11).
People are nonetheless rushing to buy properties in Singapore. A new residential condominium mission in Bedok had a long queue three days before its official launch and bought more than 350 units on the very first day. Nonetheless, there is a ‘development' that had emerged in 2011 that the majority have possibly overlooked.
In August, more than a hundred individuals across 16 states reportedly had been sickened with salmonella after consuming mangos In September, the Meals and Drug Administration detained mango imports from a Mexican packing house after the fruits had been linked to the sickenings Save Cash On Your Online Purchasing Make saving money when purchasing on-line a breeze with Bodge's favourite app, Invisible Hands. This helpful program does the price-checking analysis for you, and will warn you when a unique site has a better value - or let you realize that you have found the most effective deal. Watch Posted by Edison Foo September 29, 2013 HEARTH SALE!!! From only $9xx,000! We hope this listing will help launch your search for new properties in Singapore. It's unlawful to purchase ice cream after 6 pm, in Newark.
for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed
After obtaining a Sale License (topic to authorities circumstances meant to guard individuals shopping for property in Singapore), he might www.rmmonline.com proceed to sell units in his growth. Funding in the property market of Singapore is likely one of the few investment choices the place utilizing the bank's cash could not be any easier. The power of expats, to make a down payment, leverage the capital and consequently improve total return on investment, is excessive in Singapore. PRs who own a HDB flat must sell their flat inside six months of buying a non-public residential property in Singapore. EVERLASTING residents (PRs) now face unprecedented limits on their skill to purchase property in Singapore. New Condo in 2013 March Foreigners are eligible for Singapore greenback mortgage loan. housing grant
Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013
6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!
BuyRentSingaporeProperty.com is dedicated in providing the newest residential launches, Developer's sales, new launches, resale, HIGH tasks and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are sometimes in the vicinity of the city fringe, in areas corresponding to Newton and Novena Tanjong Rhu. These new launch projects are popular due to their proximity to town heart, the place the enterprise and entertainment centers and all the amenities they provide, but they're usually cheaper than the City core region tasks. Why submit cheques to register for new property launches in Singapore? Riversails Apartment @ Higher Serangoon LakeVille @ Jurong Lakeside Condo by MCL Ferra Condo @ Leonie Hill
If the ellipsoid is sufficiently oblate, i.e.,
b/a < ½, another class of simple closed geodesics is
possible Template:Harv. Two such geodesics
are illustrated in Figs. 13 and 14. Here
b/a = 2/7 and the equatorial azimuth,
α0, for the green (resp. blue) geodesic is chosen to
be 53.175° (resp. 75.192°), so that the geodesic completes 2
(resp. 3) complete oscillations about the equator on one circuit of the
ellipsoid.
Evaluation of the integrals
Solving the geodesic problems entails evaluating the integrals for the distance, s, and the longitude, λ, Eqs. (4) and (5). In geodetic applications where f is small, the integrals are typically evaluated as a series; for this purpose the second form of the longitude integral is preferred (since it avoids the near singular behavior of the first form when geodesics pass close to a pole). In both integrals, the integrand is an even periodic function of period π. Furthermore the term dependent on σ is multiplied by a small quantity k2 = O(f). As a consequence, the integrals can both be written in the form
where B0 = 1 + O(f) and Bj = O(f j). Series expansions for Bj can readily be found and the result truncated so that only terms which are O(f J) and larger are retained.Template:Refn (Because the longitude integral is multiplied by f, it is typically only necessary to retain terms up to O(f J−1) in that integral.) This prescription is followed by many authors Template:Harv Template:Harv Template:Harv Template:Harv Template:Harv Template:Harv. Template:Harvtxt uses J = 3 which provides an accuracy of about 0.1 mm for the WGS84 ellipsoid. Template:Harvtxt gives expansions carried out for J = 6 which suffices to provide full double precision accuracy for |f| ≤ 1/50. Trigonometric series of this type can be conveniently summed using Clenshaw summation.
In order to solve the direct geodesic problem, it is necessary to find σ given s. Since the integrand in the distance integral is positive, this problem has a unique root, which may be found using Newton's method, noting that the required derivative is just the integrand of the distance integral. Template:Harvtxt instead uses series reversion so that σ can be found without iteration; Template:Harvtxt gives a similar series.Template:Refn The reverted series converges somewhat slower that the direct series and, if |f| > 1/100, Template:Harvtxt supplements the reverted series with one step of Newton's method to maintain accuracy. Template:Harvtxt instead relies on a simpler (but slower) function iteration to solve for σ.
It is also possible to evaluate the integrals (4) and (5) by numerical quadrature Template:Harv Template:Harv Template:Harv or to apply numerical techniques for the solution of the ordinary differential equations, Eqs. (2) Template:Harv Template:Harv Template:Harv. Such techniques can be used for arbitrary flattening f. However, if f is small, e.g., |f| ≤ 1/50, they do not offer the speed and accuracy of the series expansions described above. Furthermore, for arbitrary f, the evaluation of the integrals in terms of elliptic integrals (see below) also provides a fast and accurate solution. On the other hand, Template:Harvtxt has tackled the more complex problem of geodesics on the surface at a constant altitude, h, above the ellipsoid by solving the corresponding ordinary differential equations, Eqs. (2) with [ρ, ν] replaced by [ρ + h, ν + h].
Geodesics on an ellipsoid was an early application of elliptic integrals. In particular, Template:Harvtxt writes the integrals, Eqs. (4) and (5), as
where
and
and F(φ, k), E(φ, k), and Π(φ, α2, k), are incomplete elliptic integrals in the notation of Template:Harvtxt.Template:RefnTemplate:Refn The first formula for the longitude in Eq. (7) follows directly from the first form of Eq. (5). The second formula in Eq. (7), due to Template:Harvtxt, is more convenient for calculation since the elliptic integral appears in a small term. The equivalence of the two forms follows from Template:Harvtxt. Fast algorithms for computing elliptic integrals are given by Template:Harvtxt in terms of symmetric elliptic integrals. Equation (6) is conveniently inverted using Newton's method. The use of elliptic integrals provides a good method of solving the geodesic problem for |f| > 1/50.Template:Refn
Solution of the direct problem
The basic strategy for solving the geodesic problems on the ellipsoid is to map the problem onto the auxiliary sphere by converting φ, λ, and s to β, ω and σ, solve the corresponding great-circle problem on the sphere and transfer the results back to the ellipsoid.
In implementing this program, we will frequently need to solve the "elementary" spherical triangle problem for NEP in Fig. 6 with P replaced by either A (subscript 1) or B (subscript 2). For this purpose, we can apply Napier's rules for quadrantal triangles to the triangle NEP on the auxiliary sphere which give
We can also stipulate that cosβ ≥ 0 and cosα0 ≥ 0.Template:Refn Implementing this plan for the direct problem is straightforward. We are given φ1, α1, and s12. From φ1 we obtain β1 (using the formula for the parametric latitude). We now solve the triangle problem with P = A and β1 and α1 given to find α0, σ1, and ω1.Template:Refn Use the distance and longitude equations, Eqs. (4) and (5), together with the known value of λ1, to find s1 and λ0. Determine s2 = s1 + s12 and invert the distance equation to find σ2. Solve the triangle problem with P = B and α0 and σ2 given to find β2, ω2, and α2. Convert β2 to φ2 and substitute σ2 and ω2 into the longitude equation to give λ2.
The overall method follows the procedure for solving the direct problem on a sphere. It is essentially the program laid out by Template:Harvtxt,Template:Refn Template:Harvtxt, and most subsequent authors.
Solution of the inverse problem
The ease with which the direct problem can be solved results from the fact that given φ1 and α1, we can immediately find α0, the parameter in the distance and longitude integrals, Eqs. (4) and (5). In the case of the inverse problem, we are given λ12, but we cannot easily relate this to the equivalent spherical angle ω12 because α0 is unknown. Thus, the solution of the problem requires that α0 be found iteratively. Before tackling this, it is worth understanding better the behavior of geodesics, this time, keeping the starting point fixed and varying the azimuth.
People are nonetheless rushing to buy properties in Singapore. A new residential condominium mission in Bedok had a long queue three days before its official launch and bought more than 350 units on the very first day. Nonetheless, there is a ‘development' that had emerged in 2011 that the majority have possibly overlooked.
In August, more than a hundred individuals across 16 states reportedly had been sickened with salmonella after consuming mangos In September, the Meals and Drug Administration detained mango imports from a Mexican packing house after the fruits had been linked to the sickenings Save Cash On Your Online Purchasing Make saving money when purchasing on-line a breeze with Bodge's favourite app, Invisible Hands. This helpful program does the price-checking analysis for you, and will warn you when a unique site has a better value - or let you realize that you have found the most effective deal. Watch Posted by Edison Foo September 29, 2013 HEARTH SALE!!! From only $9xx,000! We hope this listing will help launch your search for new properties in Singapore. It's unlawful to purchase ice cream after 6 pm, in Newark.
for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed
After obtaining a Sale License (topic to authorities circumstances meant to guard individuals shopping for property in Singapore), he might www.rmmonline.com proceed to sell units in his growth. Funding in the property market of Singapore is likely one of the few investment choices the place utilizing the bank's cash could not be any easier. The power of expats, to make a down payment, leverage the capital and consequently improve total return on investment, is excessive in Singapore. PRs who own a HDB flat must sell their flat inside six months of buying a non-public residential property in Singapore. EVERLASTING residents (PRs) now face unprecedented limits on their skill to purchase property in Singapore. New Condo in 2013 March Foreigners are eligible for Singapore greenback mortgage loan. housing grant
Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013
6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!
BuyRentSingaporeProperty.com is dedicated in providing the newest residential launches, Developer's sales, new launches, resale, HIGH tasks and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are sometimes in the vicinity of the city fringe, in areas corresponding to Newton and Novena Tanjong Rhu. These new launch projects are popular due to their proximity to town heart, the place the enterprise and entertainment centers and all the amenities they provide, but they're usually cheaper than the City core region tasks. Why submit cheques to register for new property launches in Singapore? Riversails Apartment @ Higher Serangoon LakeVille @ Jurong Lakeside Condo by MCL Ferra Condo @ Leonie Hill
Suppose point A in the inverse problem has φ1 = −30° and λ1 = 0°. Fig. 15 shows geodesics (in blue) emanating A with α1 a multiple of 15° up to the point at which they cease to be shortest paths. (The flattening has been increased to 1/10 in order to accentuate the ellipsoidal effects.) Also shown (in green) are curves of constant s12, which are the geodesic circles centered A. Template:Harvtxt showed that, on any surface, geodesics and geodesic circle intersect at right angles. The red line is the cut locus, the locus of points which have multiple (two in this case) shortest geodesics from A. On a sphere, the cut locus is a point. On an oblate ellipsoid (shown here), it is a segment of the circle of latitude centered on the point antipodal to A, φ = −φ1. The longitudinal extent of cut locus is approximately λ12 ∈ [π − f π cosφ1, π + f π cosφ1]. If A lies on the equator, φ1 = 0, this relation is exact and as a consequence the equator is only a shortest geodesic if |λ12| ≤ (1 − f)π. For a prolate ellipsoid, the cut locus is a segment of the anti-meridian centered on the point antipodal to A, λ12 = π, and this means that meridional geodesics stop being shortest paths before the antipodal point is reached.
The solution of the inverse problem involves determining, for a given point B with latitude φ2 and longitude λ2 which blue and green curves it lies on; this determines α1 and s12 respectively. In Fig. 16, the ellipsoid has been "rolled out" onto a plate carrée projection. Suppose φ2 = 20°, the green line in the figure. Then as α1 is varied between 0° and 180°, the longitude at which the geodesic intersects φ = φ2 varies between 0° and 180° (see Fig. 17). This behavior holds provided that |φ2| ≤ |φ1| (otherwise the geodesic does not reach φ2 for some values of α1). Thus, the inverse problem may be solved by determining the value α1 which results in the given value of λ12 when the geodesic intersects the circle φ = φ2.
This suggests the following strategy for solving the inverse problem Template:Harv. Assume that the points A and B satisfy
(There is no loss of generality in this assumption, since the symmetries of the problem can be used to generate any configuration of points from such configurations.)
- First treat the "easy" cases, geodesics which lie on a meridian or the equator. Otherwise...
- Guess a value of α1.
- Solve the so-called hybrid geodesic problem, given φ1, φ2, and α1 find λ12, s12, and α2, corresponding to the first intersection of the geodesic with the circle φ = φ2.
- Compare the resulting λ12 with the desired value and adjust α1 until the two values agree. This completes the solution.
Each of these steps requires some discussion.
1. For an oblate ellipsoid, the shortest geodesic lies on a meridian if either point lies on a pole or if λ12 = 0 or ±π. The shortest geodesic follows the equator if φ1 = φ2 = 0 and |λ12| ≤ (1 − f)π. For a prolate ellipsoid, the meridian is no longer the shortest geodesic if λ12 = ±π and the points are close to antipodal (this will be discussed in the next section). There is no longitudinal restriction on equatorial geodesics.
2. In most cases a suitable starting value of α1 is found by solving the spherical inverse problemTemplate:Refn
with ω12 = λ12. This may be a bad approximation if A and B are nearly antipodal (both the numerator and denominator in the formula above become small); however, this may not matter (depending on how step 4 is handled).
3. The solution of the hybrid geodesic problem is as follows. It starts the same way as the solution of the direct problem, solving the triangle NEP with P = A to find α0, σ1, ω1, and λ0.Template:Refn Now find α2 from sinα2 = sinα0/cosβ2, taking cosα2 ≥ 0 (corresponding to the first, northward, crossing of the circle φ = φ2). Next, σ2 is given by tanσ2 = tanβ2/cosα2 and ω2 by tanω2 = tanσ2/sinα0.Template:Refn Finally, use the distance and longitude equations, Eqs. (4) and (5), to find s12 and λ12.Template:Refn
4. In order to discuss how α1 is updated, let us define the root-finding problem in more detail. The curve in Fig. 17 shows λ12(α1; φ1, φ2) where we regard φ1 and φ2 as parameters and α1 as the independent variable. We seek the value of α1 which is the root of
where g(0) ≤ 0 and g(π) ≥ 0. In fact, there is a unique root in the interval α1 ∈ [0, π]. Any of a number of root-finding algorithms can be used to solve such an equation. Template:Harvtxt uses Newton's method, which requires a good starting guess; however it may be supplemented by a fail-safe method, such as the bisection method, to guarantee convergence.
An alternative method for solving the inverse problem is given by Template:Harvtxt. Let us rewrite the Eq. (5) as
Helmert's method entails assuming that ω12 = λ12, solving the resulting problem on auxiliary sphere, and obtaining an updated estimate of ω12 using
This process is repeated until convergence. Template:Harvtxt uses this method in his solution of the inverse problem. The drawbacks of this method are that convergence is slower than obtained using Newton's method (as described above) and, more seriously, that the process fails to converge at all for nearly antipodal points. In a subsequent report, Template:Harvtxt attempts to cure this defect; but he is only partially successful.
The shortest distance returned by the solution of the inverse problem is (obviously) uniquely defined. However, if B lies on the cut locus of A there are multiple azimuths which yield the same shortest distance. Here is a catalog of those cases:
- φ1 = −φ2 (with neither point at a pole). If α1 = α2, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by interchanging α1 and α2. (This occurs when λ12 ≈ ±π for oblate ellipsoids.)
- λ12 = ±π (with neither point at a pole). If α1 = 0 or ±π, the geodesic is unique. Otherwise there are two geodesics and the second one is obtained by negating α1 and α2. (This occurs when φ1 + φ2 ≈ 0 for prolate ellipsoids.)
- A and B are at opposite poles. There are infinitely many geodesics which can be generated by varying the azimuths so as to keep α1 + α2 constant. (For spheres, this prescription applies when A and B are antipodal.)
Differential behavior of geodesics
Various problems involving geodesics require knowing their behavior when they are perturbed. This is useful in trigonometric adjustments Template:Harv, determining the physical properties of signals which follow geodesics, etc. Consider a reference geodesic, parameterized by s the length from the northward equator crossing, and a second geodesic a small distance t(s) away from it. Template:Harvtxt showed that t(s) obeys the Gauss-Jacobi equation
where K(s) is the Gaussian curvature at s. The solution may be expressed as the sum of two independent solutions
where
We shall abbreviate m(s1, s2) = m12, the so-called reduced length, and M(s1, s2) = M12, the geodesic scale.Template:Refn Their basic definitions are illustrated in Fig. 18. Template:Harvtxt made an extensive study of their properties. The reduced length obeys a reciprocity relation,
Their derivatives are
Assuming that points 1, 2, and 3 lie on the same geodesic, then the following addition rules apply Template:Harv,
The reduced length and the geodesic scale are components of the Jacobi field.
The Gaussian curvature for an ellipsoid of revolution is
Template:Harvtxt solved the Gauss-Jacobi equation for this case obtaining
where
As we see from Fig. 18 (top sub-figure), the separation of two geodesics starting at the same point with azimuths differing by dα1 is m12 dα1. On a closed surface such as an ellipsoid, we expect m12 to oscillate about zero. Indeed, if the starting point of a geodesic is a pole, φ1 = ½π, then the reduced length is the radius of the circle of latitude, m12 = a cosβ2 = a sinσ12. Similarly, for a meridional geodesic starting on the equator, φ1 = α1 = 0, we have M12 = cosσ12. In the typical case, these quantities oscillate with a period of about 2π in σ12 and grow linearly with distance at a rate proportional to f. In trigonometric adjustments over small areas, it may be possible to approximate K(s) in Eq. (9) by a constant K. In this limit, the solutions for m12 and M12 are the same as for a sphere of radius 1/√K, namely,
To simplify the discussion of shortest paths in this paragraph we consider only geodesics with s12 > 0. The point at which m12 becomes zero is the point conjugate to the starting point. In order for a geodesic between A and B, of length s12, to be a shortest path it must satisfy the Jacobi condition Template:Harv Template:Harv Template:Harv, that there is no point conjugate to A between A and B. If this condition is not satisfied, then there is a nearby path (not necessarily a geodesic) which is shorter. Thus, the Jacobi condition is a local property of the geodesic and is only a necessary condition for the geodesic being a global shortest path. Necessary and sufficient conditions for a geodesic being the shortest path are:
- for an oblate ellipsoid, |σ12| ≤ π;
- for a prolate ellipsoid, |λ12| ≤ π, if α0 ≠ 0; if α0 = 0, the supplemental condition m12 ≥ 0 is required if |λ12| = π.
The latter condition above can be used to determine whether the shortest path is a meridian in the case of a prolate ellipsoid with |λ12| = π. The derivative required to solve the inverse method using Newton's method, ∂λ12(α1; φ1, φ2) / ∂α1, is given in terms of the reduced length Template:Harv.
Geodesic map projections
Two map projections are defined in terms of geodesics. They are based on polar and rectangular geodesic coordinates on the surface Template:Harv. The polar coordinate system (r, θ) is centered on some point A. The coordinates of another point B are given by r = s12 and θ = ½π − α1 and these coordinates are used to find the projected coordinates on a plane map, x = r cosθ and y = r sinθ. The result is the familiar azimuthal equidistant projection; in the field of the differential geometry of surfaces, it is called the exponential map. Due to the basic properties of geodesics Template:Harv, lines of constant r and lines of constant θ intersect at right angles on the surface. The scale of the projection in the radial direction is unity, while the scale in the azimuthal direction is s12/m12.
The rectangular coordinate system (x, y) uses a reference geodesic defined by A and α1 as the x axis. The point (x, y) is found by traveling a distance s13 = x from A along the reference geodesic to an intermediate point C and then turning ½π counter-clockwise and traveling along a geodesic a distance s32 = y. If A is on the equator and α1 = ½π, this gives the equidistant cylindrical projection. If α1 = 0, this gives the Cassini-Soldner projection. Cassini's map of France placed A at the Paris Observatory. Due to the basic properties of geodesics Template:Harv, lines of constant x and lines of constant y intersect at right angles on the surface. The scale of the projection in the y direction is unity, while the scale in the x direction is 1/M32.
The gnomonic projection is a projection of the sphere where all geodesics (i.e., great circles) map to straight lines (making it a convenient aid to navigation). Such a projection is only possible for surfaces of constant Gaussian curvature Template:Harv. Thus a projection in which geodesics map to straight lines is not possible for an ellipsoid. However, it is possible to construct an ellipsoidal gnomonic projection in which this property approximately holds Template:Harv. On the sphere, the gnomonic projection is the limit of a doubly azimuthal projection, a projection preserving the azimuths from two points A and B, as B approaches A. Carrying out this limit in the case of a general surface yields an azimuthal projection in which the distance from the center of projection is given by ρ = m12/M12. Even though geodesics are only approximately straight in this projection, all geodesics through the center of projection are straight. The projection can then be used to give an iterative but rapidly converging method of solving some problems involving geodesics, in particular, finding the intersection of two geodesics and finding the shortest path from a point to a geodesic.
The Hammer retroazimuthal projection is a variation of the azimuthal equidistant projection Template:Harv. A geodesic is constructed from a central point A to some other point B. The polar coordinates of the projection of B are r = s12 and θ = ½π − α2 (which depends on the azimuth at B, instead of at A). This can be used to determine the direction from an arbitrary point to some fixed center. Template:Harvtxt suggested another application: if the central point A is a beacon, such as the Rugby Clock, then at an unknown location B the range and the bearing to A can be measured and the projection can be used to estimate the location of B.
Envelope of geodesics
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for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed
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Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013
6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!
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The geodesics from a particular point A if continued past the cut locus form an envelope illustrated in Fig. 19. Here the geodesics for which α1 is a multiple of 3° are shown in light blue. (The geodesics are only shown for their first passage close to the antipodal point, not for subsequent ones.) Some geodesic circles are shown in green; these form cusps on the envelope. The cut locus is shown in red. The envelope is the locus of points which are conjugate to A; points on the envelope may be computed by finding the point at which m12 = 0 on a geodesic (and Newton's method can be used to find this point). Template:Harvtxt calls this star-like figure produced by the envelope an astroid.
Outside the astroid two geodesics intersect at each point; thus there are two geodesics (with a length approximately half the circumference of the ellipsoid) between A and these points. This corresponds to the situation on the sphere where there are "short" and "long" routes on a great circle between two points. Inside the astroid four geodesics intersect at each point. Four such geodesics are shown in Fig. 20 where the geodesics are numbered in order of increasing length. (This figure uses the same position for A as Fig. 15 and is drawn in the same projection.) The two shorter geodesics are stable, i.e., m12 > 0, so that there is no nearby path connecting the two points which is shorter; the other two are unstable. Only the shortest line (the first one) has σ12 ≤ π. All the geodesics are tangent to the envelope which is shown in green in the figure. A similar set of geodesics for the WGS84 ellipsoid is given in this table Template:Harv:
| No. | α1 (°) | α2 (°) | s12 (m) | σ12 (°) | m12 (m) |
|---|---|---|---|---|---|
| 1 | 161.890524736 | 18.090737246 | 19989832.8276 | 179.894971388 | 57277.3769 |
| 2 | 30.945226882 | 149.089121757 | 20010185.1895 | 180.116378785 | 24240.7062 |
| 3 | 68.152072881 | 111.990398904 | 20011886.5543 | 180.267429871 | −22649.2935 |
| 4 | −81.075605986 | −99.282176388 | 20049364.2525 | 180.630976969 | −68796.1679 |
The approximate shape of the astroid is given by
or, in parametric form,
The astroid is also the envelope of the family of lines
where γ is a parameter. (These are generated by the rod of the trammel of Archimedes.) This aids in finding a good starting guess for α1 for Newton's method for in inverse problem in the case of nearly antipodal points Template:Harv.
The astroid is the (exterior) evolute of the geodesic circles centered at A. Likewise the geodesic circles are involutes of the astroid.
Area of a geodesic polygon
A geodesic polygon is a polygon whose sides are geodesics. The area of such a polygon may be found by first computing the area between a geodesic segment and the equator, i.e., the area of the quadrilateral AFHB in Fig. 1 Template:Harv. Once this area is known, the area of a polygon may be computed by summing the contributions from all the edges of the polygon.
Here we develop the formula for the area S12 of AFHB following Template:Harvtxt. The area of any closed region of the ellipsoid is
where dT is an element of surface area and K is the Gaussian curvature. Now the Gauss-Bonnet theorem applied to a geodesic polygon states
where
is the geodesic excess and θj is the exterior angle at vertex j. Multiplying the equation for Γ by R22, where R2 is the authalic radius, and subtracting this from the equation for T givesTemplate:Refn
where the value of K for an ellipsoid has been substituted. Applying this formula to the quadrilateral AFHB, noting that Γ = α2 − α1, and performing the integral over φ gives
where the integral is over the geodesic line (so that φ is implicitly a function of λ). Converting this into an integral over σ, we obtain
where
and the notation E12 = α2 − α1 is used for the geodesic excess. The integral can be expressed as a series valid for small f Template:Harv Template:Harv.
The area of a geodesic polygon is given by summing S12 over its edges. This result holds provided that the polygon does not include a pole; if it does 2π R22 must be added to the sum. If the edges are specified by their vertices, then a convenient expression for E12 is
This result follows from one of Napier's analogies.
Software implementations
An implementation of Vincenty's algorithm in Fortran is provided by Template:Harvtxt. Version 3.0 includes Vincenty's treatment of nearly antipodal points Template:Harv. Vincenty's original formulas are used in many geographic information systems. Except for nearly antipodal points (where the inverse method fails to converge), this method is accurate to about 0.5 mm for the WGS84 ellipsoid.
The algorithms given in Template:Harvtxt are included in
GeographicLib Template:Harv. These are accurate to about
15 nanometers for WGS84. Implementations in several
languages (C++, C, Fortran,
Java, JavaScript,
Python, Matlab, and
Maxima) are included. In addition to solving the
basic geodesic problem, this library can return m12,
M12, M21, and S12. A
command-line utility, GeodSolve, for geodesic
calculations is included. As of version 4.9.0, the PROJ.4 library
for cartographic projections includes the C implementation. This is
exposed in the two command-line utilities, geod and
invgeod, and in the library itself. These algorithms have
also been implemented in IDL and
C#.
The solution of the geodesic problems in terms of elliptic integrals is
included in GeographicLib (in C++ only), e.g., via the -E
option to GeodSolve. This method of solution is about
2–3 time slower than using series expansions; however it
provides accurate solutions for ellipsoids of revolution with
b/a ∈ [0.01, 100] Template:Harv.
Geodesics on a triaxial ellipsoid
Solving the geodesic problem for an ellipsoid of revolution is, from the mathematical point of view, relatively simple: because of symmetry, geodesics have a constant of the motion, given by Clairaut's relation allowing the problem to be reduced to quadrature. By the early 19th century (with the work of Legendre, Oriani, Bessel, et al.), there was a complete understanding of the properties of geodesics on an ellipsoid of revolution.
On the other hand, geodesics on a triaxial ellipsoid (with 3 unequal axes) have no obvious constant of the motion and thus represented a challenging "unsolved" problem in the first half of the 19th century. In a remarkable paper, Template:Harvtxt discovered a constant of the motion allowing this problem to be reduced to quadrature also Template:Harv.Template:RefnTemplate:Refn
Triaxial coordinate systems
The key to the solution is expressing the problem in the "right" coordinate system. Consider the ellipsoid defined by
where (X,Y,Z) are Cartesian coordinates centered on the ellipsoid and, without loss of generality, a ≥ b ≥ c > 0. A point on the surface is specified by a latitude and longitude. The geographical latitude and longitude (φ, λ) are defined by
The parametric latitude and longitude (φ′, λ′) are defined by
Jacobi employed the ellipsoidal latitude and longitude (β, ω) defined by
In the limit b → a, β becomes the parametric latitude for an oblate ellipsoid, so the use of the symbol β is consistent with the previous sections. However ω is different from the spherical longitude defined above.Template:Refn
Grid lines of constant β (in blue) and ω (in green) are given in Fig. 21. In contrast to (φ, λ) and (φ′, λ′), (β, ω) is an orthogonal coordinate system: the grid lines intersect at right angles. The principal sections of the ellipsoid, defined by X = 0 and Z = 0 are shown in red. The third principal section, Y = 0, is covered by the lines β = ±90° and ω = 0° or ±180°. These lines meet at four umbilical points (two of which are visible in this figure) where the principal radii of curvature are equal. Here and in the other figures in this section the parameters of the ellipsoid are a:b:c = 1.01:1:0.8, and it is viewed in an orthographic projection from a point above φ = 40°, λ = 30°.
The grid lines of the ellipsoidal coordinates may be interpreted in three different ways
- They are "lines of curvature" on the ellipsoid, i.e., they are parallel to the directions of principal curvature Template:Harv.
- They are also intersections of the ellipsoid with confocal systems of hyperboloids of one and two sheets Template:Harv.
- Finally they are geodesic ellipses and hyperbolas defined using two adjacent umbilical points Template:Harv. For example, the lines of constant β in Fig. 21 can be generated with the familiar string construction for ellipses with the ends of the string pinned to the two umbilical points.
Conversions between these three types of latitudes and longitudes and the Cartesian coordinates are simple algebraic exercises.
The differential equations for a geodesic in ellipsoidal coordinates are
Jacobi's solution
Jacobi showed that the geodesic equations, expressed in ellipsoidal coordinates, are separable. Here is how he recounted his discovery to his friend and neighbor Bessel Template:Harv,
The day before yesterday, I reduced to quadrature the problem of geodesic lines on an ellipsoid with three unequal axes. They are the simplest formulas in the world, Abelian integrals, which become the well known elliptic integrals if 2 axes are set equal.
Königsberg, 28th Dec. '38.
The solution given by Template:Harvtxt is
As Jacobi notes "a function of the angle β equals a function of the angle ω. These two functions are just Abelian integrals..." Two constants δ and γ appear in the solution. Typically δ is zero if the lower limits of the integrals are taken to be the starting point of the geodesic and the direction of the geodesics is determined by γ. However for geodesics that start at an umbilical points, we have γ = 0 and δ determines the direction at the umbilical point. The constant γ may be expressed as
where α is the angle the geodesic makes with lines of constant ω. In the limit b → a, this reduces to sinα cosβ = const., the familiar Clairaut relation. A nice derivation of Jacobi's result is given by Template:Harvtxt where he gives the solution found by Template:Harvtxt for general quadratic surfaces. In this formulation, the distance along the geodesic, s, is found using
An alternative expression for the distance is
Survey of triaxial geodesics
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for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed
After obtaining a Sale License (topic to authorities circumstances meant to guard individuals shopping for property in Singapore), he might www.rmmonline.com proceed to sell units in his growth. Funding in the property market of Singapore is likely one of the few investment choices the place utilizing the bank's cash could not be any easier. The power of expats, to make a down payment, leverage the capital and consequently improve total return on investment, is excessive in Singapore. PRs who own a HDB flat must sell their flat inside six months of buying a non-public residential property in Singapore. EVERLASTING residents (PRs) now face unprecedented limits on their skill to purchase property in Singapore. New Condo in 2013 March Foreigners are eligible for Singapore greenback mortgage loan. housing grant
Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013
6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!
BuyRentSingaporeProperty.com is dedicated in providing the newest residential launches, Developer's sales, new launches, resale, HIGH tasks and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are sometimes in the vicinity of the city fringe, in areas corresponding to Newton and Novena Tanjong Rhu. These new launch projects are popular due to their proximity to town heart, the place the enterprise and entertainment centers and all the amenities they provide, but they're usually cheaper than the City core region tasks. Why submit cheques to register for new property launches in Singapore? Riversails Apartment @ Higher Serangoon LakeVille @ Jurong Lakeside Condo by MCL Ferra Condo @ Leonie Hill
On a triaxial ellipsoid, there are only 3 simple closed geodesics, the
three principal sections of the ellipsoid given by X = 0,
Y = 0, and Z = 0.Template:Refn
To survey the other geodesics, it is convenient to consider geodesics
which intersect the middle principal section, Y = 0, at right
angles. Such geodesics are shown in Figs. 22–26,
which use the same ellipsoid parameters and the same viewing direction
as Fig. 21. In addition, the three principal ellipses are shown
in red in each of these figures.
If the starting point is β1 ∈ (−90°, 90°), ω1 = 0, and α1 = 90°, then the geodesic encircles the ellipsoid in a "circumpolar" sense. The geodesic oscillates north and south of the equator; on each oscillation it completes slightly less that a full circuit around the ellipsoid resulting, in the typical case, in the geodesic filling the area bounded by the two latitude lines β = ±β1. Two examples are given in Figs. 22 and 23. Figure 22 shows practically the same behavior as for an oblate ellipsoid of revolution (because a ≈ b); compare to Fig. 11. However, if the starting point is at a higher latitude (Fig. 22) the distortions resulting from a ≠ b are evident. All tangents to a circumpolar geodesic touch the confocal single-sheeted hyperboloid which intersects the ellipsoid at β = β1 Template:Harv Template:Harv.
People are nonetheless rushing to buy properties in Singapore. A new residential condominium mission in Bedok had a long queue three days before its official launch and bought more than 350 units on the very first day. Nonetheless, there is a ‘development' that had emerged in 2011 that the majority have possibly overlooked.
In August, more than a hundred individuals across 16 states reportedly had been sickened with salmonella after consuming mangos In September, the Meals and Drug Administration detained mango imports from a Mexican packing house after the fruits had been linked to the sickenings Save Cash On Your Online Purchasing Make saving money when purchasing on-line a breeze with Bodge's favourite app, Invisible Hands. This helpful program does the price-checking analysis for you, and will warn you when a unique site has a better value - or let you realize that you have found the most effective deal. Watch Posted by Edison Foo September 29, 2013 HEARTH SALE!!! From only $9xx,000! We hope this listing will help launch your search for new properties in Singapore. It's unlawful to purchase ice cream after 6 pm, in Newark.
for a borrower with one or more excellent residential property loans3; and No SSD will be payable by the seller if the property is bought more than three years after it was bought. Non-extension of the Jan 2009 Funds assistance measures for the property market when the measures expire. Permitting up to 2 years of property tax deferral for land accepted for improvement. Lively secondary marketplace for easy exit of property investments. Curiosity on mortgage fee is tax deductible when the property is rented out. We've progressive property tax rates for residential properties- the higher the value of the residential property, the upper the tax charge. Why do I've to pay property tax on my house when I'm an onwer-occupier and not deriving rental earnings? Palacio Cluster Landed
After obtaining a Sale License (topic to authorities circumstances meant to guard individuals shopping for property in Singapore), he might www.rmmonline.com proceed to sell units in his growth. Funding in the property market of Singapore is likely one of the few investment choices the place utilizing the bank's cash could not be any easier. The power of expats, to make a down payment, leverage the capital and consequently improve total return on investment, is excessive in Singapore. PRs who own a HDB flat must sell their flat inside six months of buying a non-public residential property in Singapore. EVERLASTING residents (PRs) now face unprecedented limits on their skill to purchase property in Singapore. New Condo in 2013 March Foreigners are eligible for Singapore greenback mortgage loan. housing grant
Skilled property brokers in Singapore will assist the client and defend their interests throughout the acquisition, and will help safe the provide at the best possible price. The agents may even be sure that all paperwork are in order and that the vendor is the rightful proprietor of the property. Most of the property brokers share their databases with each other. Tuesday's measures come after the Monetary Authority of Singapore in June tightened rules on property loans, and closed loopholes that allowed some householders to circumvent loan restrictions and avoid paying levies when buying extra property. Contact us for a FREE housing loan analysis and to study concerning the newest mortgage to worth on Singapore housing loans. J Gateway Condominium (Bought Out) Oct, 2013
6. Completion of automotive park, roads and drains serving the housing project 5% 7. Notice of Vacant Possession (Upon obtaining Temporary Occupation Permit) The estimates given by SingaporeHousingLoan.SG's calculator are not quotations and are usually not binding in any way on SingaporeHousingLoan.sg or the consumer. The method of calculation does not exactly mirror that of which is being used by the individual monetary establishment and the data provided should be used as a information only. SingaporeHousingLoan.SG's calculator does not pre-qualify you for any housing loan from any monetary institution. Undercover Report Finds Unlawful Rat Meat Sold In London Market Unpasteurized Tempeh Linked To Salmonella Outbreak That Sickens 60 Hottest EC Launch in Sengkang!
BuyRentSingaporeProperty.com is dedicated in providing the newest residential launches, Developer's sales, new launches, resale, HIGH tasks and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are sometimes in the vicinity of the city fringe, in areas corresponding to Newton and Novena Tanjong Rhu. These new launch projects are popular due to their proximity to town heart, the place the enterprise and entertainment centers and all the amenities they provide, but they're usually cheaper than the City core region tasks. Why submit cheques to register for new property launches in Singapore? Riversails Apartment @ Higher Serangoon LakeVille @ Jurong Lakeside Condo by MCL Ferra Condo @ Leonie Hill
If the starting point is β1 = 90°,
ω1 ∈ (0°, 180°), and
α1 = 0°, then the geodesic encircles the ellipsoid
in a "transpolar" sense. The geodesic oscillates east and west of the
ellipse X = 0; on each oscillation it completes slightly more
that a full circuit around the ellipsoid resulting, in the typical case,
in the geodesic filling the area bounded by the two longitude lines
ω = ω1 and ω = 180° − ω1.
If a = b, all meridians are geodesics; the effect of
a ≠ b causes such geodesics to oscillate east and west.
Two examples are given in Figs. 24 and 25. The constriction
of the geodesic near the pole disappears in the limit
b → c; in this case, the ellipsoid becomes a
prolate ellipsoid and Fig. 24 would resemble Fig. 12
(rotated on its side). All tangents to a transpolar geodesic touch the
confocal double-sheeted hyperboloid which intersects the ellipsoid at
ω = ω1.
If the starting point is β1 = 90°, ω1 = 0° (an umbilical point), and α1 = 45° (the geodesic leaves the ellipse Y = 0 at right angles), then the geodesic repeatedly intersects the opposite umbilical point and returns to its starting point. However on each circuit the angle at which it intersects Y = 0 becomes closer to 0° or 180° so that asymptotically the geodesic lies on the ellipse Y = 0 Template:Harv Template:Harv. This is shown in Fig. 26. Note that a single geodesic does not fill an area on the ellipsoid. All tangents to umbilical geodesics touch the confocal hyperbola which intersects the ellipsoid at the umbilic points.
Umbilical geodesic enjoy several interesting properties.
- Through any point on the ellipsoid, there are two umbilical geodesics.
- The geodesic distance between opposite umbilical points is the same regardless of the initial direction of the geodesic.
- Whereas the closed geodesics on the ellipses X = 0 and Z = 0 are stable (an geodesic initially close to and nearly parallel to the ellipse remains close to the ellipse), the closed geodesic on the ellipse Y = 0, which goes through all 4 umbilical points, is exponentially unstable. If it is perturbed, it will swing out of the plane Y = 0 and flip around before returning to close to the plane. (This behavior may repeat depending on the nature of the initial perturbation.)
If the starting point A of a geodesic is not an umbilical point, then its envelope is an astroid with two cusps lying on β = −β1 and the other two on ω = ω1 + π Template:Harv. The cut locus for A is the portion of the line β = −β1 between the cusps Template:Harv.
Template:Harv gives a method for solving the inverse problem for a triaxial ellipsoid by directly integrating the system of ordinary differential equations for a geodesic. (Thus, it does not utilize Jacobi's solution.)
Applications
The direct and inverse geodesic problems no longer play the central role in geodesy that they once did. Instead of solving adjustment problems as a two-dimensional problem in spheroidal trigonometry, these problem are now solved by three-dimensional methods Template:Harv. Nevertheless, terrestrial geodesics still play an important role in several areas:
- for measuring distances and areas in geographic information systems;
- the definition of maritime boundaries Template:Harv;
- in the rules of the Federal Aviation Administration for area navigation Template:Harv;
- the method of measuring distances in the FAI Sporting Code Template:Harv.
By the principle of least action, many problems in physics can be formulated as a variational problem similar to that for geodesics. Indeed the geodesic problem is equivalent to the motion of a particle constrained to move on the surface, but otherwise subject to no forces Template:Harv Template:Harv. For this reason, geodesics on simple surfaces such as ellipsoids of revolution or triaxial ellipsoids are frequently used as "test cases" for exploring new methods. Examples include:
- the development of elliptic integrals Template:Harv and elliptic functions Template:Harv;
- the development of differential geometry Template:Harv Template:Harv;
- methods for solving systems of differential equations by a change of independent variables Template:Harv;
- the study of caustics Template:Harv;
- investigations into the number and stability of periodic orbits Template:Harv;
- in the limit c → 0, geodesics on a triaxial ellipsoid reduce to a case of dynamical billiards;
- extensions to an arbitrary number of dimensions Template:Harv;
- geodesic flow on a surface Template:Harv.
See also
Notes
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References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Template:Cite doi
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - Template:Cite doi
- Template:Cite doi
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Template:Cite doi
- One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - Template:Cite doi
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Template:Cite techreport
- One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - Template:Cite techreport
- One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Template:Cite doi
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Template:Cite doi
- Template:Cite doi
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My name is Dorris Bosanquet but everybody calls me Dorris. I'm from Iceland. I'm studying at the university (3rd year) and I play the Bass Guitar for 5 years. Usually I choose music from my famous films ;).
I have two sister. I love Knitting, watching movies and Water sports.
Also visit my blog post - http://Hostgator1Centcoupon.info/ (http://dawonls.dothome.co.kr/db/?document_srl=373691)- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Template:Cite arXiv
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Template:Cite doi
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Template:Cite techreport
- One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - Template:Cite doi
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My name is Dorris Bosanquet but everybody calls me Dorris. I'm from Iceland. I'm studying at the university (3rd year) and I play the Bass Guitar for 5 years. Usually I choose music from my famous films ;).
I have two sister. I love Knitting, watching movies and Water sports.
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - Template:Cite doi
- One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - Template:Cite doi
- Template:Cite doi
- One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - Template:Cite doi
- Template:Cite doi
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Template:Cite techreport
- Template:Cite doi
- Template:Cite doi
- Template:Cite doi
- Template:Cite doi
- Template:Cite doi
- Template:Cite doi
- Template:Cite doi
- Template:Cite techreport
- Template:Cite techreport
- One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - Template:Cite techreport
- Template:Cite techreport
- Template:Cite doi
- One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
External links
- Online geodesic bibliography, approximately 150 books and articles on geodesics on ellipsoids together with links to online copies.
- Implementations of Template:Harvtxt for oblate ellipsoids:
- NGS implementation, includes modifications described in Template:Harvtxt.
- NGS online solution of the direct problem.
- Online calculator from Geoscience Australia.
- Javascript implementations of solutions to direct problem and inverse problem.
- Implementation of Template:Harvtxt for ellipsoids of revolution in Geographiclib Template:Harv:
- GeographicLib web site for downloading library and documentation.
- GeodSolve(1), man page for a utility for geodesic calculations.
- An online version of GeodSolve.
- Planimeter(1), man page for a utility for calculating the area of geodesic polygons.
- An online version of Planimeter.
- Javascript utility for direct and inverse problems and area calculations.
- Drawing geodesics on Google Maps.
- Matlab implementation of the geodesic routines (used for the figures for geodesics on ellipsoids of revolution in this article).
- Geodesics on a triaxial ellipsoid: