projective space

Projective space and homogeneous coordinates.

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

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

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

Every $\mathbf{x}=(x_{0},\ldots,x_{n})\in\mathbb{K}^{n+1}\backslash\{0\}$ determines an element of projective space, namely the line passing through $\mathbf{x}$. Formally, this line is the equivalence class $[\mathbf{x}]$, or $[x_{0}:x_{1}:\ldots:x_{n}]$, as it is commonly denoted. The numbers $x_{0},\ldots,x_{n}$ 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 homogeneouscoordinates”.

Affine coordinates.

Projective space also admits a more conventional type of coordinate system, called affine coordinates. Let $A_{0}\subset{\mathbb{K}\mathrm{P}}^{n}$ be the subset of all elements $p=[x_{0}:x_{1}:\ldots:x_{n}]\in{\mathbb{K}\mathrm{P}}^{n}$ such that $x_{0}\neq 0$. We then define the functions

 $X_{i}:A_{0}\rightarrow\mathbb{K}^{n},\quad i=1,\ldots,n,$

according to

 $X_{i}(p)=\frac{x_{i}}{x_{0}},$

where $(x_{0},x_{1},\ldots,x_{n})$ is any element of the equivalence class representing $p$. This definition makes sense because other elements of the same equivalence class have the form

 $(y_{0},y_{1},\ldots,y_{n})=(\lambda x_{0},\lambda x_{1},\ldots,\lambda x_{n})$

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

 $\frac{y_{i}}{y_{0}}=\frac{x_{i}}{x_{0}}.$

The functions $X_{1},\ldots,X_{n}$ are called affine coordinates relative to the hyperplane

 $H_{0}=\{x_{0}=1\}\subset\mathbb{K}^{n+1}.$

Geometrically, affine coordinates can be described by saying that the elements of $A_{0}$ are lines in $\mathbb{K}^{n+1}$ that are not parallel to $H_{0}$, and that every such line intersects $H_{0}$ in one and exactly one point. Conversely points of $H_{0}$ are represented by tuples $(1,x_{1},\ldots,x_{n})$ with $(x_{1},\ldots,x_{n})\in\mathbb{K}^{n}$, and each such point uniquely labels a line $[1:x_{1}:\ldots:x_{n}]$ in $A_{0}$.

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 $\mathbb{K}^{n+1}$ that does not contain the origin. In particular, we get $n+1$ different systems of affine coordinates corresponding to the hyperplanes $\{x_{i}=1\},\;i=0,1,\ldots,n.$ 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 projectivity, is a bijective transformation of projective space that preserves all incidence relations. For $n\geq 2$, every automorphism of ${\mathbb{K}\mathrm{P}}^{n}$ is engendered by a semilinear invertible transformation of $\mathbb{K}^{n+1}$. Let $A:\mathbb{K}^{n+1}\rightarrow\mathbb{K}^{n+1}$ be an invertible semilinear transformation. The corresponding projectivity $[A]:{\mathbb{K}\mathrm{P}}^{n}\rightarrow{\mathbb{K}\mathrm{P}}^{n}$ is the transformation

 $[\mathbf{x}]\mapsto[A\mathbf{x}],\quad\mathbf{x}\in\mathbb{K}^{n+1}.$

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

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

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

A collineation 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 ${\mathrm{PGL}}_{n+1}(\mathbb{K})$ Note that for fields such as $\mathbb{R}$ and $\mathbb{C}$, the group of projective collineations is also described by the projectivizations ${\mathrm{PSL}}_{n+1}(\mathbb{R}),{\mathrm{PSL}}_{n+1}(\mathbb{C})$, of the corresponding unimodular group.

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

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

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

Title projective space ProjectiveSpace 2013-03-22 12:03:53 2013-03-22 12:03:53 rmilson (146) rmilson (146) 13 rmilson (146) Definition msc 14-00 Projectivity SemilinearTransformation homogeneous coordinates affine coordinates projective automorphism