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 (R) where Klein's quadric M 4 2 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 (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 × 4 matrices. Furthermore, we give methods to compute a versor corresponding to a given projective transformation.Mathematics Subject Classification (2010). 15A66, 51N15, 51F15, 51A05.

3D Oriented Projective Geometry Through Versors of $${\mathbb{R}^{3,3}}$$ R 3 , 3

“…Precisely what R 3,3 and its orthogonal group O(3, 3) allow is explored in Sect. 6. We find that there are useful projective transformations definable in the space of lines that do not correspond to projective transformations of points.…”

“…As a consequence, the great advantage of a versor representation (structure preservation of composition of primitives, as in CGA) appears not to materialize-in any case it is not explored in [4]. Also, the dimensionality of the representation in R • In [6], Klawitter generates the projective automorphisms of Klein's quadric in the projective space P 5 (R) by versors of the Clifford algebra R 3,3 , and relates this to the representation of the 4 × 4 homogeneous coordinate matrices of projective transformations in 3D. He shows how to convert a versor from R 3,3 to a 4 × 4 matrix, and vice versa.…”

“…The intended practical use, with its emphasis on meaningful orientation of lines, guides our treatment. As with [6], the representation of scenes in terms of lines would be most natural to R 3,3 , but requires further study to do effectively. Meanwhile, the classical scene representation in terms of points and planes can definitely be supported, since they are naturally included as 3-blades of R 3,3 .…”

“…We find that there are useful projective transformations definable in the space of lines that do not correspond to projective transformations of points. Among them are 'oriented reflections', missed in both [4] and [6]. We have no space to treat the line-containing blades fully, L. Dorst Adv.…”

“…[10]). Within oriented projective geometry we will of course not consider this projectively equivalent, by arbitrary rescaling, to a four-dimensional submanifold of a five-dimensional projective space P 5 (R), and thus we depart from the traditional treatment also followed by [6].…”