A Clifford algebraic Approach to Line Geometry

Abstract

In this paper we combine methods from projective geometry, Klein's model, and Clifford algebra. We develop a Clifford algebra whose Pin group is a double cover of the group of regular projective transformations. The Clifford algebra we use is constructed as homogeneous model for the five-dimensional real projective space $P^5(\mathbb{R})$ where Klein's quadric $M_2^4$ defines the quadratic form. We discuss all entities that can be represented naturally in this homogeneous Clifford algebra model. Projective automorphisms of Klein's quadric induce projective transformations of $P^3(\mathbb{R})$ and vice versa. Cayley-Klein geometries can be represented by Clifford algebras, where the group of Cayley-Klein isometries is given by the Pin group of the corresponding Clifford algebra. Therefore, we examine the versor group and study the correspondence between versors and regular projective transformations represented as $4\times 4$ matrices. Furthermore, we give methods to compute a versor corresponding to a given projective transformation.