projective space

Projective space and homogeneous coordinates.

Let 𝕂 be a field. Projective space of dimensionMathworldPlanetmathPlanetmathPlanetmath n over 𝕂, typically denoted by 𝕂Pn, is the set of lines passing through the origin in 𝕂n+1. More formally, consider the equivalence relationMathworldPlanetmath on the set of non-zero points 𝕂n+1\{0} defined by


Projective space is defined to be the set of the corresponding equivalence classesMathworldPlanetmath.

Every 𝐱=(x0,,xn)𝕂n+1\{0} determines an element of projective space, namely the line passing through 𝐱. Formally, this line is the equivalence class [𝐱], or [x0:x1::xn], as it is commonly denoted. The numbers x0,,xn are referred to as homogeneous coordinates of the line. Homogeneous coordinates differ from ordinary coordinate systemsMathworldPlanetmath in that a given element of projective space is labeled by multiple homogeneousPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathcoordinatesMathworldPlanetmathPlanetmath”.

Affine coordinates.

Projective space also admits a more conventional type of coordinate system, called affine coordinates. Let A0𝕂Pn be the subset of all elements p=[x0:x1::xn]𝕂Pn such that x00. We then define the functionsMathworldPlanetmath


according to


where (x0,x1,,xn) is any element of the equivalence class representing p. This definition makes sense because other elements of the same equivalence class have the form


for some non-zero λ𝕂, and hence


The functions X1,,Xn are called affine coordinates relative to the hyperplaneMathworldPlanetmathPlanetmathPlanetmath


Geometrically, affine coordinates can be described by saying that the elements of A0 are lines in 𝕂n+1 that are not parallelMathworldPlanetmathPlanetmathPlanetmath to H0, and that every such line intersects H0 in one and exactly one point. Conversely points of H0 are represented by tuples (1,x1,,xn) with (x1,,xn)𝕂n, and each such point uniquely labels a line [1:x1::xn] in A0.

It must be noted that a single system of affine coordinates does not cover all of projective space. However, it is possible to define a system of affine coordinates relative to every hyperplane in 𝕂n+1 that does not contain the origin. In particular, we get n+1 different systems of affine coordinates corresponding to the hyperplanes {xi=1},i=0,1,,n. Every element of projective space is covered by at least one of these n+1 systems of coordinates.

Projective automorphisms.

A projective automorphism, also known as a projectivityMathworldPlanetmath, is a bijectiveMathworldPlanetmathPlanetmath transformationMathworldPlanetmath of projective space that preserves all incidence relations. For n2, every automorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath of 𝕂Pn is engendered by a semilinear invertiblePlanetmathPlanetmathPlanetmath transformation of 𝕂n+1. Let A:𝕂n+1𝕂n+1 be an invertible semilinear transformation. The corresponding projectivity [A]:𝕂Pn𝕂Pn is the transformation


For every non-zero λ𝕂 the transformation λA gives the same projective automorphism as A. For this reason, it is convenient we identify the group of projective automorphisms with the quotient


Here ΓL refers to the group of invertible semi-linear transformations, while the quotienting 𝕂 refers to the subgroupMathworldPlanetmathPlanetmath of scalar multiplications.

A collineationMathworldPlanetmath is a special kind of projective automorphism, one that is engendered by a strictly linear transformation. The group of projective collineations is therefore denoted by PGLn+1(𝕂) Note that for fields such as and , the group of projective collineations is also described by the projectivizations PSLn+1(),PSLn+1(), of the corresponding unimodular groupMathworldPlanetmath.

Also note that if a field, such as , lacks non-trivial automorphisms, then all semi-linear transformations are linear. For such fields all projective automorphisms are collineations. Thus,


By contrast, since possesses non-trivial automorphisms, complex conjugation for example, the group of automorphisms of complex projective space is larger than PSLn+1(), where the latter denotes the quotient of SLn+1() by the subgroup of scalingsMathworldPlanetmath by the (n+1)st roots of unity.

Title projective space
Canonical name ProjectiveSpace
Date of creation 2013-03-22 12:03:53
Last modified on 2013-03-22 12:03:53
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 13
Author rmilson (146)
Entry type Definition
Classification msc 14-00
Related topic Projectivity
Related topic SemilinearTransformation
Defines homogeneous coordinates
Defines affine coordinates
Defines projective automorphism