Synopses & Reviews
The true power of vectors has never been exploited, for over a century, mathematicians, engineers, scientists, and more recently programmers, have been using vectors to solve an extraordinary range of problems. However, today, we can discover the true potential of oriented, lines, planes and volumes in the form of geometric algebra. As such geometric elements are central to the world of computer games and computer animation, geometric algebra offers programmers new ways of solving old problems. John Vince (best-selling author of a number of books including Geometry for Computer Graphics, Vector Analysis for Computer Graphics and Geometric Algebra for Computer Graphics) provides new insights into geometric algebra and its application to computer games and animation. The first two chapters review the products for real, complex and quaternion structures, and any non-commutative qualities that they possess. Chapter three reviews the familiar scalar and vector products and introduces the idea of 'dyadics', which provide a useful mechanism for describing the features of geometric algebra. Chapter four introduces the geometric product and defines the inner and outer products, which are employed in the following chapter on geometric algebra. Chapters six and seven cover all the 2D and 3D products between scalars, vectors, bivectors and trivectors. Chapter eight shows how geometric algebra brings new insights into reflections and rotations, especially in 3D. Finally, chapter nine explores a wide range of 2D and 3D geometric problems followed by a concluding tenth chapter. Filled with lots of clear examples, full-colour illustrations and tables, this compact book provides an excellent introduction to geometric algebra for practitioners in computer games and animation.
Geometric algebra is still treated as an obscure branch of algebra and most books have been written by competent mathematicians in a very abstract style. This restricts the readership of such books especially by programmers working in computer graphics, who simply want guidance on algorithm design. Geometric algebra provides a unified algebraic system for solving a wide variety of geometric problems. John Vince reveals the beauty of this algebraic framework and communicates to the reader new and unusual mathematical concepts using colour illustrations, tabulations, and easy-to-follow algebraic proofs. The book includes many worked examples to show how the algebra works in practice and is essential reading for anyone involved in designing 3D geometric algorithms.
This book uses 3D colour drawings and tabulations of algebraic expansions to provide new insights into geometric algebra and its application to computer games and animation. It is filled with many worked examples and full-colour illustrations and tables.
Table of Contents
Introduction.- Products.- Vector Products.- The Geometric Product.- Geometric Algebra.- Products in 2D.- Products in 3D.- Reflections and Rotations.- Applied Geomteric Algebra.- Conclusion.- Appendices.