Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry Hardcover – Apr 19 2007
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Within the last decade, Geometric Algebra (GA) has emerged as a powerful alternative to classical matrix algebra as a comprehensive conceptual language and computational system for computer science. This book will serve as a standard introduction and reference to the subject for students and experts alike. As a textbook, it provides a thorough grounding in the fundamentals of GA, with many illustrations, exercises and applications. Experts will delight in the refreshing perspective GA gives to every topic, large and small. -David Hestenes, Distinguished research Professor, Department of Physics, Arizona State University Geometric Algebra is becoming increasingly important in computer science. This book is a comprehensive introduction to Geometric Algebra with detailed descriptions of important applications. While requiring serious study, it has deep and powerful insights into GA’s usage. It has excellent discussions of how to actually implement GA on the computer. -Dr. Alyn Rockwood, CTO, FreeDesign, Inc. Longmont, Colorado--This text refers to an alternate Hardcover edition.
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The subtitle is misleading, as the authors admit in the Preface. This is a book about expressing geometric concepts algebraically, not about programming. (Nevertheless, programming examples, using C++, OpenGL, and the authors' libraries, appear throughout.) Part I (pp. 23-244) develops the main ideas of GA, including vector spaces, the geometric product, transformations, and differentiation. Part II (pp. 245-502) presents models of geometries, including vector space, homogeneous, conformal representations. Part III (pp. 503-584) discusses the software implementation of GA. The big problem is that a naive approach would be extremely inefficient, and so clever optimizations are needed. The authors' approach is to use generative programming: the user feeds parameters of the GA (dimension, metric, etc.) into a code generator, which then generates an optimized library (in C++) for that algebra.
This is a challenging book. It contains a large amount of detailed material and assumes fluency in mathematics, programming, and graphics.Read more ›
I would suggest any interested readers either one (or both) of David Estenes books on geometric algebra or Doran and Lasenby's Geometric Algebra for Physicists. The later is a very good introduction to the subject and goes in some great détails. Interested readers will then want to have a look at Estenes' books.
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It contains particularly good introductions to the dot and wedge products and how they can be applied and what they can be used to model. After one gets comfortable with these ideas they introduce the subject axiomatically. Much of the pre-axiomatic introductory material is based on the use of the scalar product, defined as a determinant. You'll have to be patient to see where and why that comes from, but this choice allows the authors to defer some of the mathematical learning overhead until one is ready for the ideas a bit better.
Having started study of the subject with papers of Hestenes, Cambridge, and Baylis papers, I found the alternate notation for the generalized dot product (L and backwards L for contraction) distracting at first but adjusting to it does not end up being that hard.
This book has three sections, the first covering the basics, the second covering the conformal applications for graphics, and the last covering implementation. As one reads geometric algebra books it is natural to wonder about this, and the pros, cons and efficiencies of various implementation techniques are discussed.
There are other web resources available associated with this book that are quite good. The best of these is GAViewer, a graphical geometric calculator that was the product of some of the research that generated this book. Performing the GAViewer tutorial exercises is a great way to build some intuition to go along with the math, putting the geometric back in the algebra.
There are specific GAViewer exercises that you can do independent of the book, and there is also an excellent interactive tutorial available. Browse the book website, or Search for '2003 Game Developer Lecture, Interactive GA tutorial. UvA GA Website: Tutorials'. Even if one decided not to learn GA, using this to play with the graphical cross product manipulation, with the ability to rotate viewpoints, is quite neat and worthwhile.
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