A Clifford algebraic Approach to Line Geometry

Abstract. In this paper we combine methods from projective geome-try, 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 homoge-neous model for the five-dimensional real projective space P5(R) where Klein’s quadric M42 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 P3(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 al-gebra. Therefore, we examine the versor group and study the corre-spondence between versors and regular projective transformations rep-resented as 4 × 4 matrices. Furthermore, we give methods to compute a versor corresponding to a given projective transformation