# 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

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

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

Every $\mathbf{x}=({x}_{0},\mathrm{\dots},{x}_{n})\in {\mathbb{K}}^{n+1}\backslash \{0\}$ determines an element of
projective space, namely the line passing through $\mathbf{x}$. Formally,
this line is the equivalence class $[\mathbf{x}]$, or $[{x}_{0}:{x}_{1}:\mathrm{\dots}:{x}_{n}]$,
as it is commonly denoted. The numbers ${x}_{0},\mathrm{\dots},{x}_{n}$ 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
$p=[{x}_{0}:{x}_{1}:\mathrm{\dots}:{x}_{n}]\in \mathbb{K}{\mathrm{P}}^{n}$ such that ${x}_{0}\ne 0$. We then
define the functions^{}

$${X}_{i}:{A}_{0}\to {\mathbb{K}}^{n},i=1,\mathrm{\dots},n,$$ |

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 $p$. This definition makes sense because other elements of the same equivalence class 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 $\lambda \in \mathbb{K}$, and hence

$$\frac{{y}_{i}}{{y}_{0}}=\frac{{x}_{i}}{{x}_{0}}.$$ |

The functions ${X}_{1},\mathrm{\dots},{X}_{n}$ are called *affine coordinates* 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
$(1,{x}_{1},\mathrm{\dots},{x}_{n})$ with $({x}_{1},\mathrm{\dots},{x}_{n})\in {\mathbb{K}}^{n}$, and each such
point 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 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 $\{{x}_{i}=1\},i=0,1,\mathrm{\dots},n.$ 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 $n\ge 2$, every automorphism^{} of $\mathbb{K}{\mathrm{P}}^{n}$ is
engendered by a semilinear invertible^{} transformation of ${\mathbb{K}}^{n+1}$.
Let $A:{\mathbb{K}}^{n+1}\to {\mathbb{K}}^{n+1}$ be an invertible semilinear
transformation. The corresponding projectivity
$[A]:\mathbb{K}{\mathrm{P}}^{n}\to \mathbb{K}{\mathrm{P}}^{n}$ is the transformation

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

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

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

Here $\mathrm{\Gamma}\mathrm{L}$ 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 $\u2102$, the group of
projective collineations is also described by the projectivizations
${\mathrm{PSL}}_{n+1}(\mathbb{R}),{\mathrm{PSL}}_{n+1}(\u2102)$, 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,

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

By contrast, since $\u2102$
possesses non-trivial automorphisms, complex conjugation for example,
the group of automorphisms of complex projective space is larger than
${\mathrm{PSL}}_{n+1}(\u2102)$, where the latter denotes the quotient of
${\mathrm{SL}}_{n+1}(\u2102)$ by the subgroup of scalings^{} 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 |