projective space

Projective space and homogeneous coordinates.

Let $𝕂$ be a field. Projective space of dimension $n$ over$𝕂$, typically denoted by $𝕂{\mathrm{P}}^n$, is the set of lines passingthrough the origin in ${𝕂}^{n+1}$. More formally, consider theequivalence relation $\sim$ on the set of non-zero points ${𝕂}^{n+1}\backslash \backslash \{0\}$defined by

$$𝐱\sim \lambda 𝐱,𝐱\in {𝕂}^{n+1}\backslash \backslash \{0\},\lambda \in 𝕂\backslash \backslash \{0\}.$$

Projective space is defined to be the set of thecorresponding equivalence classes.

Every $𝐱=(x_0,\mathrm{\dots },x_n)\in {𝕂}^{n+1}\backslash \backslash \{0\}$ determines an element ofprojective space, namely the line passing through $𝐱$. Formally,this line is the equivalence class $[𝐱]$, or $[x_0:x_1:\mathrm{\dots }:x_n]$,as it is commonly denoted. The numbers $x_0,\mathrm{\dots },x_n$ are referredto as homogeneous coordinates of the line. Homogeneous coordinatesdiffer from ordinary coordinate systems in that a given element ofprojective space is labeled by multiple homogeneous “coordinates”.

Affine coordinates.

Projective space also admits a moreconventional type of coordinate system, called affine coordinates.Let $A_0\subset 𝕂{\mathrm{P}}^n$ be the subset of all elements$p=[x_0:x_1:\mathrm{\dots }:x_n]\in 𝕂{\mathrm{P}}^n$ such that $x_0\ne 0$. We thendefine the functions

$$X_i:A_0\to {𝕂}^n,i=1,\mathrm{\dots },n,$$

according to

$$X_i(p)=\frac{x_i}{x_0},$$

where $(x_0,x_1,\mathrm{\dots },x_n)$ is any element of the equivalence class representing $p$. Thisdefinition makes sense because other elements of the same equivalenceclass have the form

$$(y_0,y_1,\mathrm{\dots },y_n)=(\lambda x_0,\lambda x_1,\mathrm{\dots },\lambda x_n)$$

for some non-zero $\lambda \in 𝕂$, and hence

$$\frac{y_i}{y_0}=\frac{x_i}{x_0}.$$

The functions $X_1,\mathrm{\dots },X_n$ are called affine coordinates relativeto the hyperplane

$$H_0=\{x_0=1\}\subset {𝕂}^{n+1}.$$

Geometrically,affine coordinates can be described by saying that the elements of$A_0$ are lines in ${𝕂}^{n+1}$ that are not parallel to $H_0$, andthat every such line intersects $H_0$ in one and exactly one point.Conversely points of $H_0$ are represented by tuples$(1,x_1,\mathrm{\dots },x_n)$ with $(x_1,\mathrm{\dots },x_n)\in {𝕂}^n$, and each suchpoint uniquely labels a line $[1:x_1:\mathrm{\dots }:x_n]$ in $A_0$.

It must be noted that a single system of affine coordinates does notcover all of projective space. However, it is possible todefine a system of affine coordinates relative to every hyperplane in${𝕂}^{n+1}$ that does not contain the origin. In particular, we get$n+1$ different systems of affine coordinates corresponding to thehyperplanes $\{x_i=1\},i=0,1,\mathrm{\dots },n.$ Every element ofprojective space is covered by at least one of these $n+1$ systems ofcoordinates.

Projective automorphisms.

A projective automorphism, also known as a projectivity, is abijective transformation of projective space that preserves allincidence relations. For $n\ge 2$, every automorphism of $𝕂{\mathrm{P}}^n$ isengendered by a semilinear invertible transformation of ${𝕂}^{n+1}$.Let $A:{𝕂}^{n+1}\to {𝕂}^{n+1}$ be an invertible semilineartransformation. The corresponding projectivity$[A]:𝕂{\mathrm{P}}^n\to 𝕂{\mathrm{P}}^n$ is the transformation

$$[𝐱]\mapsto [A𝐱],𝐱\in {𝕂}^{n+1}.$$

For every non-zero $\lambda \in 𝕂$ thetransformation $\lambda A$ gives the same projective automorphism as$A$. For this reason, it is convenient we identify the group ofprojective automorphisms with the quotient

$${\mathrm{P}\mathrm{\Gamma }\mathrm{L}}_{n+1}(𝕂)={\mathrm{\Gamma }\mathrm{L}}_{n+1}(𝕂)/𝕂.$$

Here $\mathrm{\Gamma }\mathrm{L}$ refers to the group ofinvertible semi-linear transformations, while the quotienting $𝕂$refers to the subgroup of scalar multiplications.

A collineation is a special kind of projective automorphism, one thatis engendered by a strictly linear transformation. The group ofprojective collineations is therefore denoted by ${\mathrm{PGL}}_{n+1}(𝕂)$Note that for fields such as $ℝ$ and $ℂ$, the group ofprojective collineations is also described by the projectivizations${\mathrm{PSL}}_{n+1}(ℝ),{\mathrm{PSL}}_{n+1}(ℂ)$, of the correspondingunimodular group.

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

$${\mathrm{P}\mathrm{\Gamma }\mathrm{L}}_{n+1}(ℝ)={\mathrm{PSL}}_{n+1}(ℝ)={\mathrm{SL}}_{n+1}(ℝ)/\{\pm I_{n+1}\}.$$

By contrast, since $ℂ$possesses non-trivial automorphisms, complex conjugation for example,the group of automorphisms of complex projective space is larger than${\mathrm{PSL}}_{n+1}(ℂ)$, where the latter denotes the quotient of${\mathrm{SL}}_{n+1}(ℂ)$ by the subgroup of scalings by the $(n+1)$st rootsof unity.