How to Read This Guide to Geometric Algebra in Practice

This book is called a ‘Guide to Geometric Algebra in Practice’. It is composed of chapters by experts in the field and was conceived during the AGACSE-2010 conference in Amsterdam. As you scan the contents, you will find that all chapters indeed use geometric algebra but that the term ‘practice’ means different things to different authors. As we discuss the various Parts below, we guide you through them. We will then see that appearances may deceive: some of the more theoretical looking chapters provide useful and practical techniques. This book is organized in themes of application fields, corresponding to the division into Parts. This is sometimes an arbitrary allocation; one of the powers of geometric algebra is that its unified approach permits techniques and representations from one field to be applied to another. In this guide we move, generally, from the description of physical motion and its measurement to the description of objects of a geometrical nature. Basic geometric algebra, sometimes known as Clifford algebra, is well understood and arguably has been for many years. It is important to realize that it is not just one algebra, but rather a family of algebras, all with the same essential structure. A successful application for geometric algebra involves identifying, among those in this family, an algebra that considerably facilitates a particular task at hand. The current emphasis on rigid body motion (measurement, interpolation, tracking) has focused the attention on a specific geometric algebra, the conformal model. This uses an algebra in which such motions are representable as rotations in a carefully chosen model space (for 3D, a 5D space denoted R4,1, with a 32D geometric algebra denoted R4,1). Doing so is an innovation over traditionally used representations such as homogeneous coordinates, since geometric algebra has a particularly powerful representation of rotations (as ‘rotors’—essentially spinors, with quaternions as a very special case). This conformal model (CGA, for Conformal Geometric Algebra) is used in most of this book. We provide a brief tutorial introduction to its essence in the Appendix (Chap. 21), to make this guide more self-contained, but more extensive accessible introductions may be found elsewhere. Application of geometric algebra started in physics. Using conformal geometric algebra, we can now update its description of motion in Part I: Rigid Body Motion. From there, geometric algebra migrated to applications in engineering and computer science, where motion tracking and image processing were the first to appreciate and apply its techniques. In this book these fields are represented in Part II: Interpolation and Tracking and Part III: Image Processing. More recently, traditionally