PlanetMath: projective space

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projective space

(Definition)

Projective space and homogeneous coordinates.

Let $\mathbb{K}$ be a field. Projective space of dimension

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

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

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 homogeneous “coordinates”.

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

$\displaystyle X_i:A_0\rightarrow \mathbb{K}^n,\quad i=1,\ldots,n,$

according to

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

where $(x_0,x_1,\ldots,x_n)$ is any element of the equivalence class representing

. This definition makes sense because other elements of the same equivalence class have the form

$\displaystyle (y_0,y_1,\ldots,y_n)=(\lambda x_0,\lambda x_1,\ldots,\lambda x_n)$

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

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

The functions $X_1,\ldots,X_n$ are called affine coordinates relative to the hyperplane

$\displaystyle H_0=\{x_0=1\}\subset\mathbb{K}^{n+1}.$

Geometrically, affine coordinates can be described by saying that the elements of

are lines in $\mathbb{K}^{n+1}$ that are not parallel to

, and that every such line intersects

in one and exactly one point. Conversely points of

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

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 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 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

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

For every non-zero $\lambda\in\mathbb{K}$ the transformation $\lambda A$ gives the same projective automorphism as

. For this reason, it is convenient we identify the group of projective automorphisms with the quotient

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

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,

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

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

st roots of unity.

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"projective space" is owned by rmilson. [ full author list (3) | owner history (1) ]

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Also defines:

homogeneous coordinates, affine coordinates, projective automorphism

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Cross-references: roots of unity, scalings, complex, complex conjugation, unimodular group, linear transformation, strictly, collineation, multiplications, scalar, subgroup, quotient, group, semilinear transformation, invertible, automorphism, incidence relations, preserves, transformation, bijective, projectivity, coordinates, contain, cover, labels, tuples, intersects, parallel, hyperplane, functions, subset, type, homogeneous, multiple, coordinate systems, numbers, equivalence classes, points, equivalence relation, origin, passing through, lines, dimension, field
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This is version 9 of projective space, born on 2001-12-21, modified 2008-02-12.
Object id is 1122, canonical name is ProjectiveSpace.
Accessed 14110 times total.

Classification:

AMS MSC:

14-00 (Algebraic geometry :: General reference works )

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