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.
From the reviews: "Geometric algebra (GA), a truly fascinating area of mathematics, provides a powerful, unified language of exceptional clarity and generality to describe one-, two-, three-, and higher-dimensional geometries. ... This book's outstanding feature is the use of tables and colors to develop some arithmetical details. ... The book is better suited for self-study than for the classroom. I recommend it for upper-level undergraduates, graduate students, teachers, researchers, and technical libraries." (Edgar R. Chavez, ACM Computing Reviews, December, 2009) "In the current volume, the author simplifies the presentation based on some of his new ideas on the subject. ... The volume is self-contained and can be used by students and computer graphics professionals. ... a good course in linear algebra and some mathematical maturity. Summing Up: Recommended. Computer graphics, computer animation, and computer games collections for upper-division undergraduates, graduate students, and professionals." (D. Z. Spicer, Choice, Vol. 47 (7), March, 2010) "Geometric algebra is a topic of current interest in mathematical research and in applications in physics, engineering, and computer science ... . the book is directed to a computer programming audience. ... this accessible, introductory book may convince some computer graphics programmers of the usefulness of geometric algebra ... ." (Adam Coffman, Mathematical Reviews, Issue 2011 i) "The book's true value lies in describing important geometric transformations like reflection and rotation in a systematic way, and in listing many geometric primitives ... . For people working in computer graphics or in game design, these topics could be of considerable value, and they certainly justify the book's title." (Rolf Klein, Zentralblatt MATH, Vol. 1226, 2012)
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.