|
|
| Line 1: |
Line 1: |
| {{for|the navigation on an ellipsoid|Geodesics on an ellipsoid}}
| | Hi there, I am Andrew Berryhill. Mississippi is exactly where her house is but her husband desires them to move. The preferred hobby for him and his children is fashion and he'll be beginning something else alongside with it. He is an information officer.<br><br>my homepage psychic readings ([http://bigpolis.com/blogs/post/6503 http://bigpolis.com/blogs/post/6503]) |
| | |
| '''Great-circle navigation''' is the practice of [[navigation|navigating]] a vessel (a [[ship]] or [[aircraft]]) along a [[great circle]]. A great circle track is the shortest distance between two points on the surface of a sphere; the Earth isn't exactly spherical, but the formulas for a sphere are simpler and are often accurate enough for navigation (see the [[#A numerical example|numerical example]]).
| |
| | |
| ==Course and distance==
| |
| | |
| [[File:Sphere geodesic 4sigma.svg|thumb|200px|right|Figure 1. The great circle path between (φ<sub>1</sub>, λ<sub>1</sub>) and (φ<sub>2</sub>, λ<sub>2</sub>).]]
| |
| The great circle path may be found using [[spherical trigonometry]]; this is the spherical version of the ''inverse geodesic problem''.
| |
| If a navigator begins at ''P''<sub>1</sub> = (φ<sub>1</sub>,λ<sub>1</sub>) and plans to travel the great circle to a point at point ''P''<sub>2</sub> = (φ<sub>2</sub>,λ<sub>2</sub>) (see Fig. 1, φ is the latitude, positive northward, and λ is the longitude, positive eastward), the initial and final courses α<sub>1</sub> and α<sub>2</sub> are given by [[Solution of triangles#Two sides and the included angle given|formulas for solving a spherical triangle]]
| |
| :<math>\begin{align}
| |
| \tan\alpha_1&=\frac{\sin\lambda_{12}}{ \cos\phi_1\tan\phi_2-\sin\phi_1\cos\lambda_{12}},\\
| |
| \tan\alpha_2&=\frac{\sin\lambda_{12}}{-\cos\phi_2\tan\phi_1+\sin\phi_2\cos\lambda_{12}},\\
| |
| \end{align}</math>
| |
| where λ<sub>12</sub> = λ<sub>2</sub> − λ<sub>1</sub><ref group=note>In the article on [[great-circle distance]]s,
| |
| the notation Δλ = λ<sub>12</sub>
| |
| and Δσ = σ<sub>12</sub> is used. The notation in this article is needed to
| |
| deal with differences between other points, e.g., λ<sub>01</sub>.</ref>
| |
| and the quadrants of α<sub>1</sub>,α<sub>2</sub> are determined by the signs of the numerator and denominator in the tangent formulas (e.g., using the [[atan2]] function).
| |
| The [[central angle]] between the two points, σ<sub>12</sub>, is given by
| |
| :<math>
| |
| \cos\sigma_{12}=\sin\phi_1\sin\phi_2+\cos\phi_1\cos\phi_2\cos\lambda_{12}.
| |
| </math>{{refn|group=note|If σ<sub>12</sub> is less than 0.01° (or more than 179.99°) its cosine is 0.99999999 or more, leading to some inaccuracy. To avoid this, replace the usual formula with
| |
| :<math>\tan\sigma_{12}=\frac{\sqrt{(\cos\phi_1\sin\phi_2-\sin\phi_1\cos\phi_2\cos\lambda_{12})^2 + (\cos\phi_2\sin\lambda_{12})^2}}{\sin\phi_1\sin\phi_2+\cos\phi_1\cos\phi_2\cos\lambda_{12}}.</math>
| |
| The numerator of this formula contains the quantities that were used to determine
| |
| tanα<sub>1</sub>.}}{{refn|group=note|These equations for α<sub>1</sub>,α<sub>2</sub>,σ<sub>12</sub> are suitable for implementation
| |
| on modern calculators and computers. For hand computations with logarithms,
| |
| [[Delambre]]'s analogies<ref>{{cite book
| |
| |last = Todhunter
| |
| |first = I.
| |
| |authorlink = Isaac Todhunter
| |
| |title = Spherical Trigonometry
| |
| |year = 1871
| |
| |publisher = MacMillan
| |
| |edition = 3rd
| |
| |page = 26
| |
| |url = http://books.google.com/books?id=3uBHAAAAIAAJ&pg=PA26}}
| |
| </ref> were usually used:
| |
| :<math>
| |
| \begin{align}
| |
| \cos\tfrac12(\alpha_2+\alpha_1) \sin\tfrac12\sigma_{12} &= \sin\tfrac12(\phi_2-\phi_1) \cos\tfrac12\lambda_{12},\\
| |
| \sin\tfrac12(\alpha_2+\alpha_1) \sin\tfrac12\sigma_{12} &= \cos\tfrac12(\phi_2+\phi_1) \sin\tfrac12\lambda_{12},\\
| |
| \cos\tfrac12(\alpha_2-\alpha_1) \cos\tfrac12\sigma_{12} &= \cos\tfrac12(\phi_2-\phi_1) \cos\tfrac12\lambda_{12},\\
| |
| \sin\tfrac12(\alpha_2-\alpha_1) \cos\tfrac12\sigma_{12} &= \sin\tfrac12(\phi_2+\phi_1) \sin\tfrac12\lambda_{12}.
| |
| \end{align}
| |
| </math>
| |
| McCaw<ref>{{cite journal
| |
| |last = McCaw
| |
| |first = G. T.
| |
| |title = Long lines on the Earth
| |
| |journal = Empire Survey Review
| |
| |volume = 1
| |
| |number = 6
| |
| |pages = 259–263
| |
| |year = 1932}}</ref> refers to these equations as being in "logarithmic form", by which he means
| |
| that all the trigonometric terms appear as products; this minimizes the number of table lookups
| |
| required. Furthermore, the redundancy in these formulas serves as a check in hand calculations. If using
| |
| these equations to determine the shorter path on the great circle, it is necessary to ensure
| |
| that |λ<sub>12</sub>| ≤ π (otherwise the longer path is found).}}
| |
| The distance along the great circle will then be ''s''<sub>12</sub> = ''R''σ<sub>12</sub>, where ''R'' is the assumed radius
| |
| of the earth and σ<sub>12</sub> is expressed in [[Radian#Conversions|radians]].
| |
| Using the [[Earth radius#Mean radius|mean earth radius]], ''R'' = ''R''<sub>1</sub>, yields results for
| |
| the distance ''s''<sub>12</sub> which are within 1% of the
| |
| [[Geodesics on an ellipsoid|geodesic distance]] for the [[WGS84]] ellipsoid.
| |
| | |
| ==Finding way-points==
| |
| | |
| To find the way-points, that is the positions of selected points on the great circle between
| |
| ''P''<sub>1</sub> and ''P''<sub>2</sub>, we first extrapolate the great circle back to its ''node'' ''A'', the point
| |
| at which the great circle crosses the
| |
| equator in the northward direction: let the longitude of this point be λ<sub>0</sub> — see Fig 1. The azimuth at this point, α<sub>0</sub>, is given by the spherical sine rule:
| |
| :<math>
| |
| \sin\alpha_0 = \sin\alpha_1 \cos\phi_1.
| |
| </math>{{refn|group=note|To obtain accurate results in all cases, particularly when
| |
| α<sub>0</sub> ≈ ±½π, use instead
| |
| :<math>\tan\alpha_0 = \frac
| |
| {\sin\alpha_1 \cos\phi_1}{\sqrt{\cos^2\alpha_1 + \sin^2\alpha_1\sin^2\phi_1}}.</math>}}
| |
| Let the angular distances along the great circle from ''A'' to ''P''<sub>1</sub> and ''P''<sub>2</sub> be σ<sub>01</sub> and σ<sub>02</sub> respectively. Then using [[Spherical trigonometry#Napier's rules for quadrantal triangles|Napier's rules]] we have
| |
| :<math>
| |
| \tan\sigma_{01} = \frac{\tan\phi_1}{\cos\alpha_1}
| |
| \qquad</math>(If φ<sub>1</sub> = 0 and α<sub>1</sub> = ½π, use σ<sub>01</sub> = 0).
| |
| | |
| This gives σ<sub>01</sub>, whence σ<sub>02</sub> = σ<sub>01</sub> + σ<sub>12</sub>.
| |
| | |
| The longitude at the node is found from
| |
| <!-- Don't simplify sin(sigma)/cos(sigma) to tan(sigma). This gives the wrong quadrant for lambda. -->
| |
| :<math>
| |
| \begin{align}
| |
| \tan\lambda_{01} &= \frac{\sin\alpha_0\sin\sigma_1}{\cos\sigma_1},\\
| |
| \lambda_0 &= \lambda_1 - \lambda_{01}.
| |
| \end{align}
| |
| </math>
| |
| [[File:Sphere geodesic 2sigma.svg|thumb|200px|right|Figure 2. The great circle path between a node (an equator crossing) and an arbitrary point (φ,λ).]]
| |
| Finally, calculate the position and azimuth at an arbitrary point, ''P'' (see Fig. 2), by the spherical version of the ''direct geodesic problem''.{{refn|group=note|The direct geodesic problem, finding the position of ''P''<sub>2</sub> given ''P''<sub>1</sub>, α<sub>1</sub>,
| |
| and ''s''<sub>12</sub>, can also be solved by
| |
| [[Solution of triangles#Two sides and the included angle given|formulas for solving a spherical triangle]], as follows,
| |
| :<math>
| |
| \begin{align}
| |
| \sigma_{12} &= s_{12}/R,\\
| |
| \sin\phi_2 &= \sin\phi_1\cos\sigma_{12} + \cos\phi_1\sin\sigma_{12}\cos\alpha_1,\quad\text{or}\\
| |
| \tan\phi_2 &= \frac{\sin\phi_1\cos\sigma_{12} + \cos\phi_1\sin\sigma_{12}\cos\alpha_1}
| |
| {\sqrt{ (\cos\phi_1\cos\sigma_{12} - \sin\phi_1\sin\sigma_{12}\cos\alpha_1)^2 + (\sin\sigma_{12}\sin\alpha_1)^2 }},\\
| |
| \tan\lambda_{12} &= \frac{\sin\sigma_{12}\sin\alpha_1}
| |
| {\cos\phi_1\cos\sigma_{12} - \sin\phi_1\sin\sigma_{12}\cos\alpha_1},\\
| |
| \lambda_2 &= \lambda_1 + \lambda_{12},\\
| |
| \tan\alpha_2 &= \frac{\sin\alpha_1}
| |
| {\cos\sigma_{12}\cos\alpha_1 - \tan\phi_1\sin\sigma_{12}}.
| |
| \end{align}
| |
| </math>
| |
| The solution for way-points given in the main text is more general than this solution in that
| |
| it allows | |
| way-points at specified longitudes to be found. In addition, the solution for σ
| |
| (i.e., the position of the node)
| |
| is needed when finding [[geodesics on an ellipsoid]] by means of the auxiliary sphere.}} Napier's rules give | |
| :<math>
| |
| {\color{white}.\,\qquad)}\sin\phi = \cos\alpha_0\sin\sigma,</math>{{refn|group=note|To obtain accurate results in all cases, particularly when
| |
| φ ≈ ±½π, use instead
| |
| :<math>
| |
| \tan\phi = \frac
| |
| {\cos\alpha_0\sin\sigma}{\sqrt{\cos^2\sigma + \sin^2\alpha_0\sin^2\sigma}}.</math>}}
| |
| :<!-- Don't simplify sin(sigma)/cos(sigma) to tan(sigma). This gives the wrong quadrant for lambda. --><math>
| |
| \begin{align}
| |
| \tan(\lambda - \lambda_0) &= \frac
| |
| {\sin\alpha_0\sin\sigma}{\cos\sigma},\\
| |
| \tan\alpha &= \frac
| |
| {\tan\alpha_0}{\cos\sigma}.
| |
| \end{align}
| |
| </math>
| |
| The [[atan2]] function should be used to determine
| |
| σ<sub>01</sub>,
| |
| λ, and α.
| |
| For example, to find the
| |
| midpoint of the path, substitute σ = ½(σ<sub>01</sub> + σ<sub>02</sub>); alternatively
| |
| to find the point a distance ''d'' from the starting point, take σ = σ<sub>01</sub> + ''d''/''R''.
| |
| Likewise, the ''vertex'', the point on the great
| |
| circle with greatest latitude, is found by substituting σ = +½π.
| |
| It may be convenient to parameterize the route in terms of the longitude using
| |
| :<math>\tan\phi = \cot\alpha_0\sin(\lambda-\lambda_0).</math>
| |
| Latitudes at regular intervals of longitude can be found and the resulting positions transferred to the Mercator chart
| |
| allowing the great circle to be approximated by a series of [[rhumb line]]s.
| |
| | |
| These formulas apply to a spherical model of the earth. They are also used in solving for the great circle
| |
| on the ''auxiliary sphere'' which is a device for finding the shortest path, or ''geodesic'', on
| |
| an ellipsoid of revolution; see
| |
| the article on [[geodesics on an ellipsoid]].
| |
| | |
| ==Example==
| |
| | |
| Compute the great circle route from [[Valparaíso]],
| |
| φ<sub>1</sub> = −33°,
| |
| λ<sub>1</sub> = −71.6°, to
| |
| [[Shanghai]],
| |
| φ<sub>2</sub> = 31.4°,
| |
| λ<sub>2</sub> = 121.8°.
| |
| | |
| The formulas for course and distance give
| |
| λ<sub>12</sub> = −166.6°,
| |
| α<sub>1</sub> = −94.41°,
| |
| α<sub>2</sub> = −78.42°, and
| |
| σ<sub>12</sub> = 168.56°. Taking the [[Earth radius#Mean radius|earth radius]] to be
| |
| ''R'' = 6371 km, the distance is
| |
| ''s''<sub>12</sub> = 18743 km.
| |
| | |
| To compute points along the route, first find
| |
| α<sub>0</sub> = −56.74°,
| |
| σ<sub>1</sub> = −96.76°,
| |
| σ<sub>2</sub> = 71.8°,
| |
| λ<sub>01</sub> = 98.07°, and
| |
| λ<sub>0</sub> = −169.67°.
| |
| Then to compute the midpoint of the route (for example), take
| |
| σ = ½(σ<sub>1</sub> + σ<sub>2</sub>) = −12.48°, and solve
| |
| for
| |
| φ = −6.81°,
| |
| λ = −159.18°, and
| |
| α = −57.36°.
| |
| | |
| If the geodesic is computed accurately on the [[WGS84]] ellipsoid,<ref>
| |
| {{cite journal
| |
| |first=C. F. F.
| |
| |last=Karney
| |
| |title=Algorithms for geodesics
| |
| |journal=J. Geodesy
| |
| |volume = 87
| |
| |number = 1
| |
| |year = 2013
| |
| |pages = 43–55
| |
| |accessdate=2012-07-14
| |
| |doi=10.1007/s00190-012-0578-z
| |
| |url=http://dx.doi.org/10.1007/s00190-012-0578-z
| |
| }}
| |
| </ref> the results
| |
| are α<sub>1</sub> = −94.82°, α<sub>2</sub> = −78.29°, and
| |
| ''s''<sub>12</sub> = 18752 km. The midpoint of the geodesic is
| |
| φ = −7.07°, λ = −159.31°,
| |
| α = −57.45°.
| |
| | |
| ==Gnomonic chart==
| |
| A straight line drawn on a [[Map projection#Gnomonic|Gnomonic chart]] would be a great circle track. When this is transferred to a [[Map projection#Mercator|Mercator]] chart, it becomes a curve. The positions are transferred at a convenient interval of [[longitude]] and this is plotted on the Mercator chart.
| |
| | |
| ==See also==
| |
| * [[Great circle]]
| |
| * [[Great-circle distance]]
| |
| * [[Rhumb line]]
| |
| * [[Geographical distance]]
| |
| * [[Spherical Trigonometry]]
| |
| * [[Geodesics on an ellipsoid]]
| |
| | |
| ==Notes==
| |
| {{reflist|group=note}}
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| ==Resources==
| |
| * [http://www.greatcirclemapper.net The Great Circle Mapper] Displays Great Circle flight routes on a Map And calculates distance and duration
| |
| * [http://mathworld.wolfram.com/GreatCircle.html Great Circle – from MathWorld] Great Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999
| |
| * [http://www.gcmap.com/ Great Circle Mapper] Interactive tool for plotting great circle routes.
| |
| * [http://williams.best.vwh.net/gccalc.htm Great Circle Calculator] deriving (initial) course and distance between two points.
| |
| * [http://www.acscdg.com/ Great Circle Distance] Graphical tool for drawing great circles over maps. Also shows distance and azimuth in a table.
| |
| | |
| [[Category:Navigation]]
| |