Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics, 1st Edition (Hardback) book cover

Understanding Geometric Algebra

Hamilton, Grassmann, and Clifford for Computer Vision and Graphics, 1st Edition

By Kenichi Kanatani

A K Peters/CRC Press

208 pages | 74 B/W Illus.

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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.

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

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.

About the Author

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.

Subject Categories

BISAC Subject Codes/Headings:
COM012000
COMPUTERS / Computer Graphics
MAT002000
MATHEMATICS / Algebra / General
TEC037000
TECHNOLOGY & ENGINEERING / Robotics