projective space (original) (raw)

Projective space and homogeneous coordinates.

Every 𝐱=(x0,…,xn)∈𝕂n+1\{0} determines an element ofprojective 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 systems in that a given element of projective space is labeled by multiple homogeneous “coordinates”.

Affine coordinates.

Projective space also admits a more conventional type of coordinate system, called affine coordinates. Let A0⊂𝕂⁢Pn be the subset of all elements p=[x0:x1:…:xn]∈𝕂Pn such that x0≠0. We then define the functions

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 λ∈𝕂, and hence

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

Geometrically, affine coordinates can be described by saying that the elements ofA0 are lines in 𝕂n+1 that are not parallel 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 in𝕂n+1 that does not contain the origin. In particular, we getn+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 collineation is a special kind of projective automorphism, one that is engendered by a strictly linear transformation. The group ofprojective collineations is therefore denoted by PGLn+1⁢(𝕂)Note that for fields such as ℝ and ℂ, the group of projective collineations is also described by the projectivizationsPSLn+1⁢(ℝ),PSLn+1⁢(ℂ), of the correspondingunimodular group.

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

By contrast, since ℂpossesses non-trivial automorphisms, complex conjugation for example, the group of automorphisms of complex projective space is larger thanPSLn+1⁢(ℂ), where the latter denotes the quotient ofSLn+1⁢(ℂ) by the subgroup of scalings by the (n+1)st roots of unity.