Projective space and homogeneous coordinates.
Let be a field. Projective space of dimension over , typically denoted by , is the set of lines passing through the origin in . More formally, consider the equivalence relation on the set of non-zero points defined by
Projective space is defined to be the set of the corresponding equivalence classes.
Every determines an element of projective space, namely the line passing through . Formally, this line is the equivalence class , or , as it is commonly denoted. The numbers 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”.
where is any element of the equivalence class representing . This definition makes sense because other elements of the same equivalence class have the form
for some non-zero , and hence
The functions are called affine coordinates relative to the hyperplane
Geometrically, affine coordinates can be described by saying that the elements of are lines in that are not parallel to , and that every such line intersects in one and exactly one point. Conversely points of are represented by tuples with , and each such point uniquely labels a line 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 that does not contain the origin. In particular, we get different systems of affine coordinates corresponding to the hyperplanes Every element of projective space is covered by at least one of these systems of coordinates.
A projective automorphism, also known as a projectivity, is a bijective transformation of projective space that preserves all incidence relations. For , every automorphism of is engendered by a semilinear invertible transformation of . Let be an invertible semilinear transformation. The corresponding projectivity is the transformation
For every non-zero the transformation gives the same projective automorphism as . For this reason, it is convenient we identify the group of projective automorphisms with the quotient
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 Note that for fields such as and , the group of projective collineations is also described by the projectivizations , of the corresponding unimodular group.
Also note that if a field, such as , lacks non-trivial automorphisms, then all semi-linear transformations are linear. For such fields all projective automorphisms are collineations. Thus,
By contrast, since possesses non-trivial automorphisms, complex conjugation for example, the group of automorphisms of complex projective space is larger than , where the latter denotes the quotient of by the subgroup of scalings by the st roots of unity.
|Date of creation||2013-03-22 12:03:53|
|Last modified on||2013-03-22 12:03:53|
|Last modified by||rmilson (146)|