1st Edition

Understanding Geometric Algebra
Hamilton, Grassmann, and Clifford for Computer Vision and Graphics




ISBN 9781482259506
Published April 6, 2015 by A K Peters/CRC Press
208 Pages - 74 B/W Illustrations

USD $94.95

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Book Description

Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford. It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision.

Unlike similar texts, this book first gives separate descriptions of the various algebras and then explains how they are combined to define the field of geometric algebra. It starts with 3D Euclidean geometry along with discussions as to how the descriptions of geometry could be altered if using a non-orthogonal (oblique) coordinate system. The text focuses on Hamilton’s quaternion algebra, Grassmann’s outer product algebra, and Clifford algebra that underlies the mathematical structure of geometric algebra. It also presents points and lines in 3D as objects in 4D in the projective geometry framework; explores conformal geometry in 5D, which is the main ingredient of geometric algebra; and delves into the mathematical analysis of camera imaging geometry involving circles and spheres.

With useful historical notes and exercises, this book gives readers insight into the mathematical theories behind complicated geometric computations. It helps readers understand the foundation of today’s geometric algebra.

Table of Contents

Introduction
PURPOSE OF THIS BOOK
ORGANIZATION OF THIS BOOK
OTHER FEATURES

3D Euclidean Geometry
VECTORS
BASIS AND COMPONENTS
INNER PRODUCT AND NORM
VECTOR PRODUCTS
SCALAR TRIPLE PRODUCT
PROJECTION, REJECTION, AND REFLECTION
ROTATION
PLANES
LINES
PLANES AND LINES

Oblique Coordinate Systems
RECIPROCAL BASIS
RECIPROCAL COMPONENTS
INNER, VECTOR, AND SCALAR TRIPLE PRODUCTS
METRIC TENSOR
RECIPROCITY OF EXPRESSIONS
COORDINATE TRANSFORMATIONS

Hamilton’s Quaternion Algebra
QUATERNIONS
ALGEBRA OF QUATERNIONS
CONJUGATE, NORM, AND INVERSE
REPRESENTATION OF ROTATION BY QUATERNION

Grassmann’s Outer Product Algebra
SUBSPACES
OUTER PRODUCT ALGEBRA
CONTRACTION
NORM
DUALITY
DIRECT AND DUAL REPRESENTATIONS

Geometric Product and Clifford Algebra
GRASSMANN ALGEBRA OF MULTIVECTORS
CLIFFORD ALGEBRA
PARITY OF MULTIVECTORS
GRASSMANN ALGEBRA IN THE CLIFFORD ALGEBRA
PROPERTIES OF THE GEOMETRIC PRODUCT
PROJECTION, REJECTION, AND REFLECTION
ROTATION AND GEOMETRIC PRODUCT
VERSORS

Homogeneous Space and Grassmann-Cayley Algebra
HOMOGENEOUS SPACE
POINTS AT INFINITY
PLUCKER COORDINATES OF LINES
PLUCKER COORDINATES OF PLANES
DUAL REPRESENTATION
DUALITY THEOREM

Conformal Space and Conformal Geometry: Geometric Algebra
CONFORMAL SPACE AND INNER PRODUCT
REPRESENTATION OF POINTS, PLANES, AND SPHERES
GRASSMANN ALGEBRA IN CONFORMAL SPACE
DUAL REPRESENTATION
CLIFFORD ALGEBRA IN THE CONFORMAL SPACE
CONFORMAL GEOMETRY

Camera Imaging and Conformal Transformations
PERSPECTIVE CAMERAS
FISHEYE LENS CAMERAS
OMNIDIRECTIONAL CAMERAS
3D ANALYSIS OF OMNIDIRECTIONAL IMAGES
OMNIDIRECTIONAL CAMERAS WITH HYPERBOLIC AND ELLIPTIC MIRRORS

Answers

Bibliography

Index

Supplemental Notes and Exercises appear at the end of each chapter.

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Author(s)

Biography

Kenichi Kanatani is a professor emeritus at Okayama University. A fellow of IEICE and IEEE, Dr. Kanatani is the author of numerous books on computer vision and applied mathematics. He is also a board member of several journals and conferences.

Reviews

"Several software tools are available for executing geometric algebra, but the purpose of the book is to bring about a deeper insight and interest in the theory on which these tools are based."
Zentralblatt MATH 1319