Projective coordinate system

A projective coordinate system allows the position of a point in a projective space to be clearly described by specifying a coordinate vector. In this way, in the mathematical fields of geometry and linear algebra, the structure-preserving images of projective spaces (that is, the collineations and above all the projective images ) can be represented by coordinate- related mapping matrices and the spaces can be examined using methods of analytical geometry .

The components of the coordinate vector that describes a point in projective space are called projective coordinates . They are also known as homogeneous coordinates . (→ The main article " Homogeneous coordinates " also explains how projective coordinates can be used to identify elements of related structures such as affine spaces .)

In an abstract projective space of finite dimensions , the coordinate system is determined by suitably chosen base points - the points must be chosen in a general position and are then referred to as the projective base . The reference to the base points, instead of a vector space base (Hamel basis) , which is fully sufficient in the standard model, enables a model-independent geometrical description of the reference system and in the synthetic geometry of the introduction of comparable coordinates also in general structures (particularly projective incidence plane ) to which no vector space and thereby no body can be assigned as a coordinate area. ${\displaystyle n}$ ${\displaystyle n+2}$

Projective coordinates in the standard model

Let it be the -dimensional projective space above the body . ${\displaystyle KP^{n}}$ ${\displaystyle n}$ ${\displaystyle K}$

The projective points that belong to a vector space basis des , that is, the one-dimensional subspaces generated by these basis vectors ${\displaystyle {\mathcal {B}}=({\vec {e}}_{0},{\vec {e}}_{1},\ldots ,{\vec {e}}_{n})}$ ${\displaystyle K^{n+1}}$

${\displaystyle B_{j}=\lbrace r\cdot {\vec {e_{j}}}:\;r\in K\rbrace ;\quad 0\leq j\leq n}$

form together with the unit point

${\displaystyle E=B_{n+1}=\lbrace r\cdot \left({\vec {e}}_{0}+{\vec {e}}_{1}+\cdots {\vec {e}}_{n}\right):\;r\in K\rbrace }$

a projective basis (also: projective point basis) of the projective space . ${\displaystyle {\mathcal {B}}_{p}=(B_{0},B_{1},\ldots ,B_{n},B_{n+1})}$ ${\displaystyle KP^{n}}$

An affine space is obtained by slitting along the projective hyperplane running through . In this is the zero point. We consider for the intersection of the straight line with the hyperplane through . These points form an affine base of with the zero point . With this base we can define affine coordinates in and the projective coordinates with respect to the chosen projective base are then by definition . ${\displaystyle B_{1},\ldots ,B_{n}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle E}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle E_{i}}$ ${\displaystyle EB_{i}}$ ${\displaystyle B_{0},\ldots ,B_{i-1},B_{i+1},\ldots ,B_{n}}$ ${\displaystyle \left\{E_{1},\ldots ,E_{n}\right\}}$ ${\displaystyle E}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (x_{1},\ldots ,x_{n})}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (1;x_{1};\ldots ;x_{n})}$

example

It will be the room with the standard base ${\displaystyle KP^{2}}$

${\displaystyle B_{0}=\left[1:0:0\right],B_{1}=\left[0:1:0\right],B_{2}=\left[0:0:1\right],B_{3}=\left[1:1:1\right]}$

considered. Then have the projective straight lines

${\displaystyle B_{3}B_{1}=\left\{\left[1:1+s:1\right]\mid s\in K\right\}\cup \left\{B_{1}\right\}}$ and ${\displaystyle B_{0}B_{2}=\left\{\left[t:0:1-t\right]\mid t\in K\right\}}$

the intersection and the projective straight lines ${\displaystyle E_{1}=\left[1:0:1\right]}$

${\displaystyle B_{3}B_{2}=\left\{\left[1:1:1+s\right]\mid s\in K\right\}\cup \left\{B_{2}\right\}}$ and ${\displaystyle B_{0}B_{1}=\left\{\left[t:1-t:0\right]\mid t\in K\right\}}$

the intersection . The projective coordinates of the point are then for . ${\displaystyle E_{2}=\left[1:1:0\right]}$ ${\displaystyle \left[x:y:z\right]}$ ${\displaystyle (1;{\tfrac {y}{x}};{\tfrac {z}{x}})}$ ${\displaystyle x\not =0}$

Projective coordinates in synthetic geometry

A projective point base (red) determines a clear affine point base (green), whereby the straight line connecting it becomes the long line. ${\displaystyle (B_{0},B_{1},B_{2},E)}$ ${\displaystyle (O=B_{0},E_{1},E_{2})}$ ${\displaystyle u=B_{1}B_{2}}$

In any desired, even non-Desargue projective plane, projective coordinates can be introduced with the help of affine coordinates after a projective base has been selected .

In the projective plane, a projective base must first have been selected, that is, no three of the four points should lie on a common straight line. The point becomes the origin of the affine coordinate system, the connecting line to its first, to its second coordinate axis. The initially still projective intersection points and are the unit points on these axes, thus an affine point base is the affine plane, which arises from the projective plane by slitting along the straight line . This straight line becomes the distance line of the affine plane, see also the figure on the right. ${\displaystyle (B_{0},B_{1},B_{2},E)}$ ${\displaystyle B_{0}}$ ${\displaystyle O=B_{0}}$ ${\displaystyle B_{0}B_{1}}$ ${\displaystyle B_{0}B_{2}}$ ${\displaystyle E_{1}=EB_{2}\cap OB_{1}}$ ${\displaystyle E_{2}=EB_{1}\cap OB_{2}}$ ${\displaystyle (O,E_{1},E_{2})}$ ${\displaystyle u=B_{1}B_{2}}$

• For each point of the slotted plane , affine coordinates can be determined by coordinate construction, the coordinate range being represented by the first axis of the affine coordinate system. → The coordinate construction is described in the article Ternary body . ${\displaystyle (x_{1};x_{2})\in K^{2}}$ ${\displaystyle K}$
• A point outside of with the affine coordinates receives the projective coordinates . ${\displaystyle u}$ ${\displaystyle (x_{1};x_{2})}$ ${\displaystyle (1;x_{1};x_{2})}$
• A point on the long line receives the projective coordinates , the affine coordinates of the point being on the connecting line . (From the assumed "general situation" it follows and therefore .) ${\displaystyle U}$ ${\displaystyle u}$ ${\displaystyle (0;x_{1};x_{2})}$ ${\displaystyle (x_{1};x_{2})}$ ${\displaystyle U}$ ${\displaystyle OU}$ ${\displaystyle O\not \in u}$ ${\displaystyle U\not =O}$

The coordinates determined in this way are unique for points outside of ; for points on this uniqueness can be achieved through additional agreements. In general, they are not homogeneous: In the coordinate area , which is a ternary body, no “scalar multiplication” can generally be defined. ${\displaystyle u}$ ${\displaystyle u}$ ${\displaystyle K}$

Applications

Illustrations

If and are projective spaces of dimension or over a solid body , then: ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle n}$ ${\displaystyle m}$ ${\displaystyle K}$

• Each projective transformation of after having fixed relative to the selected point projective bases and a representation . The mapping matrix has rows and columns and is uniquely determined except for a scalar factor . ${\displaystyle \pi }$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle \pi \colon \left[x_{0},\ldots ,x_{n}\right]^{T}\rightarrow \left[(A\cdot (x_{0},\ldots ,x_{n})^{T})\right]^{T}}$ ${\displaystyle A}$ ${\displaystyle n+1}$ ${\displaystyle m+1}$ ${\displaystyle r\in K\setminus \lbrace {0}\rbrace }$
• If we choose at any point of a projective point-based or equivalent to points in general position, each an arbitrary pixel , then this can be clearly to a projective mapping continue , in which therefore applies to each base point. ${\displaystyle B_{j}(0\leq j\leq n+1)}$ ${\displaystyle P}$ ${\displaystyle n+2}$ ${\displaystyle C_{j}\in Q}$ ${\displaystyle \pi \colon P\rightarrow Q}$ ${\displaystyle \pi (B_{j})=C_{j}}$
• Each projectivity in respect has a fixed base selected projective point in a representation . The square, regular mapping matrix is uniquely determined except for a scalar factor . ${\displaystyle \pi }$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle \pi \colon \left[x_{0},\ldots ,x_{n}\right]^{T}\rightarrow \left[(A\cdot (x_{0},\ldots ,x_{n})^{T})\right]^{T}}$ ${\displaystyle (n+1)\times (n+1)}$ ${\displaystyle A}$ ${\displaystyle r\in K\setminus \lbrace {0}\rbrace }$
• For archetype points in general position and image points in general position there exists a projectivity on where is. One therefore also says that the projective linear group operates sharply, simply transitive, on the set of -tuples of points in general position. ${\displaystyle n+2}$ ${\displaystyle B_{j}(0\leq j\leq n+1)}$ ${\displaystyle n+2}$ ${\displaystyle C_{j}(0\leq j\leq n+1)}$ ${\displaystyle \pi }$ ${\displaystyle P}$ ${\displaystyle \pi (B_{j})=C_{j},(0\leq j\leq n+1)}$ ${\displaystyle \operatorname {PGL} (n+1,K)}$ ${\displaystyle n+2}$
• If the dimension is , then each collineation can be represented on a firmly selected projective basis as a composition with a projectivity and an automorphism of the body . ${\displaystyle n\geq 2}$ ${\displaystyle \kappa }$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle \kappa =\pi \circ \sigma }$ ${\displaystyle \pi }$ ${\displaystyle \sigma }$ ${\displaystyle K}$

Double ratio

The double ratio of four collinear points in a projective space is the simple ratio of the projective coordinates that the point has when the other three points are chosen as the point base of the common straight line. Here are the base points and the unit point of the coordinate system. Now with respect to this system, the coordinate representation , then for the cross ratio: . This relationship is one of the reasons that the double ratio is sometimes referred to as the inhomogeneous projective coordinate of (with regard to the other points in the double ratio). ${\displaystyle P,Q,R,S}$ ${\displaystyle P}$ ${\displaystyle B_{0}=R,\,B_{1}=S}$ ${\displaystyle E=B_{2}=Q}$ ${\displaystyle P}$ ${\displaystyle P=\left[p_{0};p_{1}\right]}$ ${\displaystyle t=\operatorname {DV} (PQRS)={\tfrac {p_{1}}{p_{0}}}}$ ${\displaystyle t\in K\cup \lbrace \infty \rbrace }$ ${\displaystyle P}$

Parametric equations

• The straight line connecting two different points and has the homogeneous parametric representation ${\displaystyle A=\left[a_{0};a_{1};\ldots a_{n}\right]}$ ${\displaystyle B=\left[b_{0};b_{1};\ldots b_{n}\right]}$

${\displaystyle \langle A,B\rangle :\;{\vec {x}}=\alpha \cdot {\begin{pmatrix}a_{0}\\a_{1}\\\vdots \\a_{n}\end{pmatrix}}+\beta \cdot {\begin{pmatrix}b_{0}\\b_{1}\\\vdots \\b_{n}\end{pmatrix}}}$

There are then for the projective coordinates of a straight line point ${\displaystyle {\vec {x}}}$ ${\displaystyle (\alpha ;\beta )\in K^{2}\setminus \lbrace 0\rbrace }$ ${\displaystyle X=\left[{\vec {x}}^{\,T}\right]}$

• More generally, the connecting space of points , whose coordinate vectors are linearly independent, is a -dimensional subspace of the projective space with the parametric representation ${\displaystyle k}$ ${\displaystyle A_{j}=\left[{{\vec {a}}_{j}}^{T}\right];\;1\leq j\leq k}$ ${\displaystyle k-1}$

${\displaystyle \langle A_{1},A_{2},\ldots A_{k}\rangle :{\vec {x}}=\sum _{j=1}^{k}\alpha _{j}{\vec {a}}_{j};\quad (\alpha _{1},\alpha _{2},\ldots \alpha _{k})\in K^{k}\setminus \lbrace 0\rbrace .}$

Coordinate equations and hyperplane coordinates

After choosing a projective point base in a -dimensional projective space , each point can be clearly assigned the coordinate equation, the set of solutions, understood as point coordinates, describing a -dimensional subspace of , i.e. a hyperplane . Since the equation is homogeneous, its solution set does not change if you multiply each coordinate by the same scalar , so the hyperplane only depends on the point and the chosen projective coordinate system. The coordinate vector is called the hyperplane coordinates of this hyperplane. A hyperplane is uniquely assigned to each point of space through dualization . ${\displaystyle {\mathcal {B}}_{p}}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle P=\left[p_{0};p_{1};\ldots p_{n}\right]}$ ${\displaystyle p_{0}\cdot x_{0}+p_{1}\cdot x_{1}+\cdots +p_{n}\cdot x_{n}=0\;}$ ${\displaystyle n-1}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle r\in K^{*}}$ ${\displaystyle P}$ ${\displaystyle P^{D}=\left[p_{0};p_{1};\ldots p_{n}\right]^{D}}$ ${\displaystyle P\rightarrow P^{D}}$

Duality in projective spaces

The dual assignment of points to hyperplanes can be expanded to a duality in the association of the projective sub-spaces of a projective space. The following assignments apply:

term	Dual term
Point	Hyperplane
Total space	Empty set as a -dimensional subspace ${\displaystyle -1}$
${\displaystyle k}$-dimensional subspace	${\displaystyle n-1-k}$-dimensional subspace
Section of two sub-spaces ${\displaystyle S\cap T}$	Connecting space of two sub-spaces ${\displaystyle S^{D}\vee T^{D}}$
Double ratio of four collinear points	Double ratio of four hyperplanes that intersect in a -dimensional subspace ${\displaystyle n-2}$

The assignment is also to be understood the other way round, since the dualization is involutive: a point corresponds to a hyperplane. While the concrete dualization depends on the chosen coordinate system, general propositions are not affected.

The duality principle of projective geometry is based on the algebraic dual space of the finite-dimensional coordinate vector space , see the main article " Dual space ". Examples of application in plane geometry can be found in " Duality (mathematics) " in the section " Principle of duality in projective geometry and in incidence structures ". ${\displaystyle K^{n+1}}$

Three-dimensional examples

In a three-dimensional space , the set of straight lines (a straight line corresponds to a 2-dimensional subspace of ) is dual to itself. The concrete straight line ${\displaystyle KP^{3}}$ ${\displaystyle K^{4}}$

${\displaystyle g=\langle e_{0},e_{1}\rangle =\lbrace \left[r,s,0,0\right]:\;(r,s)\in K^{2}\setminus \lbrace 0\rbrace \rbrace }$

is dual to

${\displaystyle g^{D}=\lbrace \left[x_{0},x_{1},x_{2},x_{3}\right]:(x_{0},x_{1},x_{2},x_{3})\in K^{4}\setminus \lbrace 0\rbrace ,x_{0}=x_{1}=0\rbrace =\langle e_{2},e_{3}\rangle }$

This is a crooked line! The statement "The straight lines and do not intersect." Is dual to "The connection space of and the entire three-dimensional space." For any two skew lines and is always a spot basis are chosen with regard to the true - to choose at any straight two linearly independent, generating vectors and supplements these four vectors with their sum as a unit point. So the statements “two straight lines do not intersect” and “two straight lines span the space” are dual descriptions of the property “skewed”. ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle g^{D}}$ ${\displaystyle g^{D}}$ ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g^{D}=h}$

On the other hand, the statements " and intersect at a point" and " and span a plane" are equivalent but not dual to one another, since the first statement does not apply to any pairs of straight lines and the dual statement about other straight lines! ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle h}$

literature

• Harold Scott MacDonald Coxeter : Real projective geometry of the plane , Munich 1955
• Hermann Schaal: Linear Algebra and Analytical Geometry, Volume II , Vieweg 1980, ISBN 3-528-13057-1
• Günter Pickert : Projective levels. 2nd edition, Berlin-Heidelberg-New York 1975

Individual evidence

1. ^ Hermann Schaal: Lineare Algebra und Analytische Geometrie, Volume II , p. 153, Vieweg 1980, ISBN 3-528-13057-1