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Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry [Hardcover]

Leo Dorst , Daniel Fontijne , Stephen Mann
5.0 out of 5 stars  See all reviews (1 customer review)
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Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry 5.0 out of 5 stars (1)
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Book Description

April 19 2007 0123694655 978-0123694652 1

Until recently, almost all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex-often a lot of effort is required to bring about even modest performance enhancements. Although linear algebra is an efficient way to specify low-level computations, it is not a suitable high-level language for geometric programming.

Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhancing way to represent the geometry of 3D objects in computer programs. In this book you will find an introduction to GA that will give you a strong grasp of its relationship to linear algebra and its significance for your work. You will learn how to use GA to represent objects and perform geometric operations on them. And you will begin mastering proven techniques for making GA an integral part of your applications in a way that simplifies your code without slowing it down.

* The first book on Geometric Algebra for programmers in computer graphics and entertainment computing

* Written by leaders in the field providing essential information on this new technique for 3D graphics

* This full colour book includes a website with GAViewer, a program to experiment with GA

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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.

Book Description

The first book on a new technique in 3D graphics

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1 of 1 people found the following review helpful
5.0 out of 5 stars A new start for computer graphics June 16 2010
The more familiar we are with a particular technique, the harder it is to see its limitations. For those who have taken the appropriate courses, the machinery of real and complex numbers, vectors, matrices, and so on, seem to be the inevitable "right" way of doing things. Yet there is another way, pioneered in the nineteenth century by Grassmann (1844) and Clifford (1878) and called (by Clifford) "geometric algebra" (GA). For over a century, GA was buried by the "standard" approach of Hamilton, Gibbs, et al., but it has recently been resurrected by a small but dedicated group of people, perhaps most notably by David Hestenes in "New Foundations for Classical Mechanics" (1999).

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.
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Amazon.com: 4.5 out of 5 stars  4 reviews
39 of 41 people found the following review helpful
5.0 out of 5 stars A reader from Los Alamos, NM Aug. 17 2007
By J. Hanlon - Published on Amazon.com
Geometric Algebra (GA) is a unifying mathematical language that should be taught instead of or at least in combination with traditional vector analysis. Most other books on GA are aimed at Physicists. This book is a better match for Engineers and Programmers. The authors are all active researchers in applications of GA. They have done a comprehensive and up to date job of collecting, organizing and presenting the material for both beginners and those who follow the development of GA on the web. The examples and problems use GAViewer, an easy to learn programming language with an Open GL view window that can be downloaded for free from the book website. Using GAViewer with the book is very good way to learn GA, especially the 5D Conformal model of 3D space. The authors hold nothing back. Between the book, the code and the website everything is there to make learning GA fun and useful. I highly recommend this book.
12 of 12 people found the following review helpful
5.0 out of 5 stars An excellent introduction to the subject. Sept. 5 2009
By Peeter Joot - Published on Amazon.com
The book Geometric Algebra For Computer Science, by Dorst, Fontijne, and Mann has one of the best introductions to the subject that I have seen.

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.
3 of 4 people found the following review helpful
5.0 out of 5 stars very good text May 10 2010
By T. Czyczko - Published on Amazon.com
This is the text I would first recommend to anyone involved in geometrical programming who would like to learn geometrical algebra.
4 of 13 people found the following review helpful
3.0 out of 5 stars ok, but... Oct. 5 2010
By Fuga Federico - Published on Amazon.com
Format:Hardcover|Verified Purchase
It's a good book, but the mathematics is poorly treated, not enough rigorous as would be expected.
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