Definitions and general properties
Recall that collineations send any three collinear points to three collinear points. Thus if a collineation fixes more than a hyperplane of points then it in fact fixes all the points of the geometry and so it is the identity map. Therefore a central collineation can be viewed the simplest of the non-identity collineations.
Suppose that a central collineation is not the identity. Then the hyperplane of fixed points is unique and receives the title of the axis of the central collineation. There is one further important result which justifies the name “central”.
Given a non-identity central collineation , there is a unique point such that for all other points , it follows that , and are collinear.
Central collineations in coordinates
Suppose we have a projective geometry of dimension , that is, we exclude now the case of projective lines and planes. The the geometry can be coordinatized through so that we may regard the projective geometry as the lattice of subspaces of a vector space of dimension over a division ring . Following the fundamental theorem of projective geometry we further know that every collineation is induced by a semi-linear transformation of . So it is possible to explore central collineations as semi-linear transformations.
Every hyperplane is a kernel of some linear functional, so we let be a linear functional of with . Furthermore, we fix so that (which implieas also that ). Hence, for each , where and .
Let such that induces a central collineation on with axis . As every scalar multiple of induces the same collineation of , we may assume that is the identity on . Using the decomposition given by we have
Suppose instead that is any linear functional of . Then select some such that . Then
fixes all the points of so induces a central collineation.
If we wish to do the same without appealing to linear functionals, we may select a basis such that and . As is selected to be the identity on we have so far specified by the matrix:
in the basis .
|Date of creation||2013-03-22 16:03:13|
|Last modified on||2013-03-22 16:03:13|
|Last modified by||Algeboy (12884)|