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

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.

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 NORMDUALITY 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 PRODUCTPROJECTION, 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 AlgebraCONFORMAL 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.

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.