central collineations

Definitions and general properties

Definition 1.

A collineationMathworldPlanetmath of a finite dimensional projective geometryMathworldPlanetmath is a central collineation if there is a hyperplaneMathworldPlanetmathPlanetmath of points fixed by the collineation.

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 geometryMathworldPlanetmath and so it is the identity mapMathworldPlanetmath. Therefore a central collineation can be viewed the simplest of the non-identity collineations.

Theorem 2.

Every collineation of a finite dimensional projective geometry of dimensionPlanetmathPlanetmath n>1 is a productPlanetmathPlanetmath of at most n central collineations. In particular, the automorphism groupMathworldPlanetmath of a projective geometry of dimension n>1 is generated by central collineations.

Suppose that a central collineation is not the identityPlanetmathPlanetmathPlanetmath. 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”.

Proposition 3.

Given a non-identity central collineation f, there is a unique point C such that for all other points P, it follows that C, P and Pf are collinearMathworldPlanetmath.

The point C determined by PropositionPlanetmathPlanetmath 3 is called the center of the non-identity central collineation. It is possible for the center to lie on the axis.

Central collineations in coordinates

Suppose we have a projective geometry of dimension n>2, 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 latticeMathworldPlanetmath of subspacesPlanetmathPlanetmath of a vector spaceMathworldPlanetmath V of dimension n+1 over a division ring Δ. Following the fundamental theorem of projective geometryMathworldPlanetmath we further know that every collineation is induced by a semi-linear transformation of V. So it is possible to explore central collineations as semi-linear transformations.

Every hyperplane is a kernel of some linear functionalPlanetmathPlanetmathPlanetmath, so we let φ:VΔ be a linear functional of V with Hφ=0. Furthermore, we fix vV so that vφ=1 (which implieas also that vH). Hence, for each uV, u=(u-(uφ)v)+(uφ)v where u-(uφ)vH and (uφ)vv.

Let fGLΔ(V) such that f induces a central collineation f~ on PG(V) with axis HV. As every scalar multiple of f induces the same collineation of PG(V), we may assume that f is the identity on H. Using the decomposition given by φ we have




Suppose instead that φ is any linear functional of V. Then select some v^V such that v^φ-1. Then


fixes all the points of kerφ so g induces a central collineation.

If we wish to do the same without appealing to linear functionals, we may select a basis {v1,,vn+1} such that H=v1,,vn and vn+1φ=1. As f is selected to be the identity on H we have so far specified f by the matrix:


in the basis {v1,,vn,vn+1}.

Title central collineations
Canonical name CentralCollineations
Date of creation 2013-03-22 16:03:13
Last modified on 2013-03-22 16:03:13
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 8
Author Algeboy (12884)
Entry type Definition
Classification msc 51A10
Classification msc 51A05
Related topic PerspectivityMathworldPlanetmath
Defines transvection
Defines center
Defines axis
Defines central collineation