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Über dieses Buch

After revising known representations of the group of Euclidean displacements Daniel Klawitter gives a comprehensive introduction into Clifford algebras. The Clifford algebra calculus is used to construct new models that allow descriptions of the group of projective transformations and inversions with respect to hyperquadrics. Afterwards, chain geometries over Clifford algebras and their subchain geometries are examined. The author applies this theory and the developed methods to the homogeneous Clifford algebra model corresponding to Euclidean geometry. Moreover, kinematic mappings for special Cayley-Klein geometries are developed. These mappings allow a description of existing kinematic mappings in a unifying framework.



1. Models and Representations

In this chapter we give a survey on methods that are used nowadays to describe spatial displacements and other transformations. Therefore, point models for lines and spatial displacements in three-dimensional Euclidean space are presented. Moreover, different methods to describe displacements are examined. We review old concepts such as dual numbers, quaternions, and dual quaternions. Furthermore, we give a short review about homogeneous coordinates and how to ” linearize ” displacements. After that we consider Clifford algebras and special Clifford algebra models. These models are a current field of research.
Daniel Klawitter

2. Chain Geometry over Clifford Algebras

In the following chapter we give an introduction to chain geometry. It is not the aim of this work to give a complete treatise of this topic, we just introduce the concepts we need for our purposes. For a more detailed introduction the reader is referred to [11]. The roots of chain geometry can be found in BENZ [5]. BENZ investigated projective lines over commutative two-dimensional algebras and the corresponding chain geometries. A more recent treatise is [33].
Daniel Klawitter

3. Kinematic Mappings for Spin Groups

In this chapter we define Cayley-Klein spaces and show how to describe certain Cayley-Klein spaces within the homogeneous Clifford algebra model. This construction is accomplished in detail for the threedimensional Euclidean space. Furthermore, the kinematic mapping of Study and the mapping of Blaschke and Grünwald are constructed in a unified method. Matrices of the collineations in the image and pre-image space are derived. The construction is accomplished in detail for the Euclidean spaces of dimension two and three. After that, we give an overview of possible kinematic mappings for Cayley-Klein spaces of dimension two and three. Moreover, the mapping for the four-dimensional Euclidean space is presented. This chapter is already published, see [41].
Daniel Klawitter


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