Global index grammar

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. In Euclidean, non-projective geometry, just as two (distinct) points determine a line (a degree-1 plane curve), five points determine a conic (a degree-2 plane curve). There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines. In Projective geometry, a line is defined via three points, all families of circles are conics passing through two points of a line at infinity, and coordinates are the basic conic objects of study, and are defined as all conics passing through the line and two points at infinity.

Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-degenerate; this is true over both the affine plane and projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion.

Proofs

This result can be proven numerous different ways; the dimension counting argument is most direct, and generalizes to higher degree, while other proofs are special to conics.

Dimension counting

Intuitively, passing through five points in general linear position specifies five independent linear constraints on the (projective) linear space of conics, and hence specifies a unique conic, though this brief statement ignores subtleties.

More precisely, this is seen as follows:

• conics correspond to points in the five-dimensional projective space ${\mathbf{P}}^5;$
• requiring a conic to pass through a point imposes a linear condition on the coordinates: for a fixed $(x,y),$ the equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$ is a linear equation in $(A,B,C,D,E,F);$
• by dimension counting five constraints (passing through five points) is necessary to specify a conic, as each constraints cuts the dimension of possibilities by 1, and one starts with 5 dimensions;
• in 5 dimensions, the intersection of 5 (independent) hyperplanes is a single point (formally, by Bézout's theorem);
• general linear position of the points means that the constraints are independent, and thus do specify a unique conic;
• the resulting conic is non-degenerate because it is a curve (since it has more than 1 point), and does not contain a line (else it would split as two lines, at least one of which must contain 3 of the 5 points, by the pigeonhole principle), so it is irreducible.

The two subtleties in the above analysis are that the resulting point is a quadratic equation (not a linear equation), and that the constraints are independent. The first is simple: if A, B, and C all vanish, then the equation $Dx+Ey+F=0$ defines a line, and any 3 points on this (indeed any number of points) lie on a line – thus general linear position ensures a conic. The second, that the constraints are independent, is significantly subtler: it corresponds to the fact that given five points in general linear position in the plane, their images in ${\mathbf{P}}^5$ under the Veronese map are in general linear position, which is true because the Veronese map is biregular: i.e., if the image of five points satisfy a relation, then the relation can be pulled back and the original points must also satisfy a relation. The Veronese map has coordinates $[x^2:xy:y^2:xz:yz:z^2],$ and the target ${\mathbf{P}}^5$ is dual to the $[A:B:C:D:E:F]$ ${\mathbf{P}}^5$ of conics. The Veronese map corresponds to "evaluation of a conic at a point", and the statement about independence of constraints is exactly a geometric statement about this map.

Synthetic proof

That five points determine a conic can be proven by synthetic geometry—i.e., in terms of lines and points in the plane—in addition to the analytic (algebraic) proof given above. Such a proof can be given using a theorem of Jakob Steiner, which states:

Given a projective transformation f, between the pencil of lines passing through a point X and the pencil of lines passing through a point Y, the set C of intersection points between a line x and its image $f(x)$ forms a conic.

Note that X and Y are on this conic by considering the preimage and image of the line XY (which is respectively a line through X and a line through Y).

This can be shown by taking the points X and Y to the standard points $[1:0:0]$ and $[0:1:0]$ by a projective transformation, in which case the pencils of lines correspond to the horizontal and vertical lines in the plane, and the intersections of corresponding lines to the graph of a function, which (must be shown) is a hyperbola, hence a conic, hence the original curve C is a conic.

Now given five points X, Y, A, B, C, the three lines $XA,XB,XC$ can be taken to the three lines $YA,YB,YC$ by a unique projective transform, since projective transforms are simply 3-transitive on lines (they are simply 3-transitive on points, hence by projective duality they are 3-transitive on lines). Under this map A maps to B, since these are the unique intersection points of these lines, and thus satisfy the hypothesis of Steiner’s theorem. The resulting conic thus contains all five points, and is the unique such conic, as desired.

Parabola construction, given five points

Construction

Given five points, one can construct the conic containing them in various ways.

Analytically, the equation for the conic can be found by linear algebra, by writing and solving the five equations in the coefficients: five equations, six unknowns, but homogeneous so scaling removes one dimension; concretely, setting one of the coefficients to 1 accomplishes this.

Synthetically, the conic can be constructed by the Template:Visible anchor, by applying the Braikenridge–Maclaurin theorem, which is the converse of Pascal's theorem. Pascal's theorem states that given 6 points on a conic (a hexagon), the lines defined by opposite sides intersect in three collinear points. This can be reversed to construct the possible locations for a 6th point, given 5 existing ones.

Generalizations

The natural generalization is to ask for what value of k a configuration of k points (in general position) in n-space determines a variety of degree d and dimension m, which is a fundamental question in enumerative geometry.

A simple case of this is for a hypersurface (a codimension 1 subvariety, the zeros of a single polynomial, the case $m=n-1$), of which plane curves are an example.

In the case of a hypersurface, the answer is given in terms of the multiset coefficient, more familiarly the binomial coefficient, or more elegantly the rising factorial, as:

$k=\left((\genfrac{}{}{0ex}{}{n+1}{d})\right)-1=(\genfrac{}{}{0ex}{}{n+d}{d})-1=\frac{1}{n!}(d+1{)}^{(n)}-1.$

This is via the analogous analysis of the Veronese map: k points in general position impose k independent linear conditions on a variety (because the Veronese map is biregular), and the number of monomials of degree d in $n+1$ variables (n-dimensional projective space has $n+1$ homogeneous coordinates) is $\left((\genfrac{}{}{0ex}{}{n+1}{d})\right),$ from which 1 is subtracted because of projectivization: multiplying a polynomial by a constant does not change its zeros.

In the above formula, the number of points k is a polynomial in d of degree n, with leading coefficient $1/n!$

In the case of plane curves, where $n=2,$ the formula becomes:

$\frac{1}{2}(d+1)(d+2)-1=\frac{1}{2}(d^2+3d)$

whose values for $d=0,1,2,3,4$ are $0,2,5,9,14$ – there are no curves of degree 0 (a single point is determines a point, which is codimension 2), 2 points determine a line, 5 points determine a conic, 9 points determine a cubic, 14 points determine a quartic, and so forth.

Related results

While five points determine a conic, sets of six or more points on a conic are in special position, as demonstrated in Pascal's theorem.

Conversely, four points do not determine a conic, but rather a pencil, the 1-dimensional linear system of conics which all pass through the four points (formally, have the four points as base locus). Similarly, three points determine a 2-dimensional linear system (net), two points determine a 3-dimensional linear system (web), one point determines a 4-dimensional linear system, and zero points place no constraints on the 5-dimensional linear system of all conics.

The Apollonian circles are two 1-parameter families determined by 2 points.

Fewer points are required to determine more specific conics – three points determine a circle, while two points determine a pencil of circles, as in the Apollonian circles.

Similarly, while nine points determine a cubic, if the nine points lie on more than one cubic—i.e., they are the intersection of two cubics—then they are not in general position, and indeed satisfy an addition constraint, as stated in the Cayley–Bacharach theorem.

Tangency

Instead of passing through points, a different condition on a curve is being tangent to a given line. Being tangent to five given lines also determines a conic, by projective duality, but from the algebraic point of view tangency to a line is a quadratic constraint, so naive dimension counting yields 2⁵ = 32 conics tangent to five given lines, of which 31 must be ascribed to degenerate conics, as described in fudge factors in enumerative geometry; formalizing this intuition requires significant further development to justify.

Another classic problem in enumerative geometry, of similar vintage to conics, is the Problem of Apollonius: a circle that is tangent to three circles in general determines eight circles, as each of these is a quadratic condition and 2³ = 8. As a question in real geometry, a full analysis involves many special cases, and the actual number of circles may be any number between 0 and 8, except for 7.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro. Template:Refbegin

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  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
  
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• 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

External links

• Five Points Determine a Conic Section, Wolfram interactive demonstration