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Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry (Morgan Kaufmann Series in Computer Graphics)

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Synopses & Reviews

Publisher Comments:

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

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.

Features

-Explains GA as a natural extension of linear algebra and conveys its significance for 3D programming of geometry in graphics, vision, and robotics.

-Systematically explores the concepts and techniques that are key to representing elementary objects and geometric operators using GA.

-Covers in detail the conformal model, a convenient way to implement 3D geometry using a 5D representation space.

-Presents effective approaches to making GA an integral part of your programming.

-Includes numerous drills and programming exercises helpful for both students and practitioners.

-Companion web site includes links to GAViewer, a program that will allow you to interact with many of the 3D figures in the book, and Gaigen 2, the platform for the instructive programming exercises that conclude each chapter.

About the Authors

Leo Dorst is Assistant Professor of Computer Science at the University of Amsterdam, where his research focuses on geometrical issues in robotics and computer vision. He earned M.Sc. and Ph.D. degrees from Delft University of Technology and received a NYIPLA Inventor of the Year award in 2005 for his work in robot path planning.

Daniel Fontijne holds a Master’s degree in artificial Intelligence and is a Ph.D. candidate in Computer Science at the University of Amsterdam. His main professional interests are computer graphics, motion capture, and computer vision.

Stephen Mann is Associate Professor in the David R. Cheriton School of Computer Science at the University of Waterloo, where his research focuses on geometric modeling and computer graphics. He has a B.A. in Computer Science and Pure Mathematics from the University of California, Berkeley, and a Ph.D. in Computer Science and Engineering from the University of Washington.

Synopsis:

Until recently, 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.

  • Explains GA as a natural extension of linear algebra and conveys its significance for 3D programming of geometry in graphics, vision, and robotics.
  • Systematically explores the concepts and techniques that are key to representing elementary objects and geometric operators using GA.
  • Covers in detail the conformal model, a convenient way to implement 3D geometry using a 5D representation space.
  • Presents effective approaches to making GA an integral part of your programming.
  • Includes numerous drills and programming exercises helpful for both students and practitioners.
  • Companion web site includes links to GAViewer, a program that will allow you to interact with many of the 3D figures in the book, and Gaigen 2, the platform for the instructive programming exercises that conclude each chapter.

Synopsis:

Proven techniques for making Geometric Algebra an integral part of your applications in a way that simplifies your code without slowing it down.

About the Author

Daniel Fontijne holds a Master’s degree in artificial Intelligence and a Ph.D. in Computer Science, both from the University of Amsterdam. His main professional interests are computer graphics, motion capture, and computer vision.

University of Waterloo, Ontario, Canada

Table of Contents

CHAPTER 1. WHY GEOMETRIC ALGEBRA?

PART I GEOMETRIC ALGEBRA

CHAPTER 2. SPANNING ORIENTED SUBSPACES

CHAPTER 3. METRIC PRODUCTS OF SUBSPACES

CHAPTER 4. LINEAR TRANSFORMATIONS OF

SUBSPACES

CHAPTER 5. INTERSECTION AND UNION OF

SUBSPACES

CHAPTER 6. THE FUNDAMENTAL PRODUCT OF

GEOMETRIC ALGEBRA

CHAPTER 7. ORTHOGONAL TRANSFORMATIONS AS

VERSORS

CHAPTER 8. GEOMETRIC DIFFERENTIATION

PART II MODELS OF GEOMETRIES

CHAPTER 9. MODELING GEOMETRIES

CHAPTER 10. THE VECTOR SPACE MODEL: THE

ALGEBRA OF DIRECTIONS

CHAPTER 11. THE HOMOGENEOUS MODEL

CHAPTER 12. APPLICATIONS OF THE

HOMOGENEOUS MODEL

CHAPTER 13. THE CONFORMAL MODEL:

OPERATIONAL EUCLIDEAN GEOMETRY

CHAPTER 14. NEW PRIMITIVES FOR EUCLIDEAN

GEOMETRY

CHAPTER 15. CONSTRUCTIONS IN EUCLIDEAN

GEOMETRY

CHAPTER 16. CONFORMAL OPERATORS

CHAPTER 17. OPERATIONAL MODELS FOR

GEOMETRIES

PART III IMPLEMENTING GEOMETRIC ALGEBRA

CHAPTER 18. IMPLEMENTATION ISSUES

CHAPTER 19. BASIS BLADES AND OPERATIONS

CHAPTER 20. THE LINEAR PRODUCTS AND

OPERATIONS

CHAPTER 21. FUNDAMENTAL ALGORITHMS FOR

NONLINEAR PRODUCTS

CHAPTER 22. SPECIALIZING THE STRUCTURE FOR

EFFICIENCY

CHAPTER 23. USING THE GEOMETRY IN A RAY-

TRACING APPLICATION

PART IV APPENDICES

A METRICS AND NULL VECTORS

B CONTRACTIONS AND OTHER INNER PRODUCTS

C SUBSPACE PRODUCTS RETRIEVED

D COMMON EQUATIONS

BIBLIOGRAPHY

INDEX

Product Details

ISBN:
9780123749420
Author:
Dorst, Leo
Publisher:
Morgan Kaufmann Publishers
Author:
Fontijne, Daniel
Author:
Mann, Stephen
Subject:
Computer graphics
Subject:
Object-oriented methods (Computer science)
Subject:
Geometry - Algebraic
Subject:
Computer Graphics - General
Subject:
Graphics-General
Series:
Morgan Kaufmann Series in Computer Graphics
Publication Date:
20090331
Binding:
HARDCOVER
Language:
English
Illustrations:
Y
Pages:
664
Dimensions:
9.45 x 7.75 in

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Related Subjects

Computers and Internet » Graphics » General
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Geometric Algebra for Computer Science (Revised Edition): An Object-Oriented Approach to Geometry (Morgan Kaufmann Series in Computer Graphics) New Hardcover
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$109.25 Backorder
Product details 664 pages Morgan Kaufmann Publishers - English 9780123749420 Reviews:
"Synopsis" by , Until recently, 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.

  • Explains GA as a natural extension of linear algebra and conveys its significance for 3D programming of geometry in graphics, vision, and robotics.
  • Systematically explores the concepts and techniques that are key to representing elementary objects and geometric operators using GA.
  • Covers in detail the conformal model, a convenient way to implement 3D geometry using a 5D representation space.
  • Presents effective approaches to making GA an integral part of your programming.
  • Includes numerous drills and programming exercises helpful for both students and practitioners.
  • Companion web site includes links to GAViewer, a program that will allow you to interact with many of the 3D figures in the book, and Gaigen 2, the platform for the instructive programming exercises that conclude each chapter.

"Synopsis" by , Proven techniques for making Geometric Algebra an integral part of your applications in a way that simplifies your code without slowing it down.
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