Geometric Algebra for Computer Graphics

Since its invention, geometric algebra has been applied to various branches of physics such as cosmology and electrodynamics, and is now being embraced by the computer graphics community where it is providing new ways of solving geometric problems. It took over two thousand years to discover this algebra, which uses a simple and consistent notation to describe vectors and their products. John Vince (best-selling author of a number of books including Geometry for Computer Graphics and Vector Analysis for Computer Graphics) tackles this new subject in his usual inimitable style, and provides an accessible and very readable introduction. The first five chapters review the algebras of real numbers, complex numbers, vectors, and quaternions and their associated axioms, together with the geometric conventions employed in analytical geometry. As well as putting geometric algebra into its historical context, John Vince provides chapters on Grassmann's outer product and Clifford's geometric product, followed by the application of geometric algebra to reflections, rotations, lines, planes and their intersection. The conformal model is also covered, where a 5D Minkowski space provides an unusual platform for unifying the transforms associated with 3D Euclidean space. Filled with lots of clear examples and useful illustrations, this compact book provides an excellent introduction to geometric algebra for computer graphics. In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions In Recent Years The Methods Of Modern Differential Geometry Have Become Of Considerable Importance In Theoretical Physics And Have Found Application In Relativity And Cosmology, High-energy Physics And Field Theory, Thermodynamics, Fluid Dynamics And Mechanics. This Textbook Provides An Introduction To These Methods - In Particular Lie Derivatives, Lie Groups And Differential Forms - And Covers Their Extensive Applications To Theoretical Physics. The Reader Is Assumed To Have Some Familiarity With Advanced Calculus, Linear Algebra And A Little Elementary Operator Theory.--cover. 1. Some Basic Mathematics -- 2. Differentiable Manifolds And Tensors -- 3. Lie Derivatives And Lie Groups -- 4. Differential Forms -- 5. Applications In Physics -- 6. Connections For Riemannian Manifolds And Gauge Theories. Bernard F. Schutz. Includes Bibliographical References And Index. Geometric algebra (a Clifford Algebra) has been applied to different branches of physics for a long time but is now being adopted by the computer graphics community and is providing exciting new ways of solving 3D geometric problems. The author tackles this complex subject with inimitable style, and provides an accessible and very readable introduction. The book is filled with lots of clear examples and is very well illustrated. Introductory chapters look at algebraic axioms, vector algebra and geometric conventions and the book closes with a chapter on how the algebra is applied to computer graphics. Front Matter....Pages i-xvi Introduction....Pages 1-3 Elementary Algebra....Pages 5-10 Complex Algebra....Pages 11-22 Vector Algebra....Pages 23-37 Quaternion Algebra....Pages 39-48 Geometric Conventions....Pages 49-54 Geometric Algebra....Pages 55-77 The Geometric Product....Pages 79-124 Reflections and Rotations....Pages 125-153 Geometric Algebra and Geometry....Pages 155-197 Conformal Geometry....Pages 199-230 Applications of Geometric Algebra....Pages 231-240 Programming Tools for Geometric Algebra....Pages 241-242 Conclusion....Pages 243-243 Back Matter....Pages 245-252