Geometric computations with Clifford algebras
TABLE OF CONTENTS PART I A UNI ED ALGEBRAIC APPROACH FOR CLASSICAL GEOMETRIES 1 NEW ALGEBRAIC TOOLS FOR CLASSICAL GEOMETRY 3 1 1 INTRODUCTION 3 1 2 GEOMETRIC ALGEBRA OF A VECTOR SPACE 4 1 3 LINEAR TRANSFORMATIONS 13 1 4 VECTORS AS GEOMETRICAL POINTS 19 1 5 LINEARIZING THE EUCLIDEAN GROUP 23 2 GENERALIZED HOMOGENEOUS COORDINATES FOR COMPUTATIONAL GEOMETRY 27 2 1 INTRODUCTION 27 2 2 MINKOWSKI SPACE WITH CONFORMAL AND PROJECTIVE SPLITS 29 2 3 HOMOGENEOUS MODEL OF EUCLIDEAN SPACE 33 2 4 EUCLIDEAN SPHERES AND HYPERSPHERES 40 2 5 MULTI DIMENSIONAL SPHERES PLANES AND SIMPLEXES 41 2 6 RELATION AMONG SPHERES AND HYPERPLANES 46 2 7 CONFORMAL TRANSFORMATIONS 52 3 SPHERICAL CONFORMAL GEOMETRY WITH GEOMETRIC ALGEBRA 61 3 1 INTRODUCTION 61 3 2 HOMOGENEOUS MODEL OF SPHERICAL SPACE 62 3 3 RELATION BETWEEN TWO SPHERES OR HYPERPLANES 66 3 4 SPHERES AND PLANES OF DIMENSION R 68 3 5 STEREOGRAPHIC PROJECTION 70 3 6 CONFORMAL TRANSFORMATIONS 72 3 6 1 INVERSIONS AND RE ECTIONS 72 3 6 2 OTHER TYPICAL CONFORMAL TRANSFORMATIONS 73 XII TABLE OF CONTENTS 4 A UNIVERSAL MODEL FOR CONFORMAL GEOMETRIES OF EUCLIDEAN SPHERICAL AND DOUBLE HYPERBOLIC SPACES 77 4 1 INTRODUCTION 77 4 2 THE HYPERBOLOID MODEL 79 4 2 1 GENERALIZED POINTS 79 4 2 2 TOTAL SPHERES 83 4 3 THE HOMOGENEOUS MODEL 83 4 3 1 GENERALIZED POINTS 84 4 3 2 TOTAL SPHERES 85 4 3 3 TOTAL SPHERES OF DIMENSIONAL R 89 4 4 STEREOGRAPHIC PROJECTION 90 4 5 THE CONFORMAL BALL MODEL 92 4 6 THE HEMISPHERE MODEL 93 4 7 THE HALF SPACE MODEL 94 4 8 THE KLEIN BALL MODEL 97 4 9 A UNIVERSAL MODEL FOR EUCLIDEAN SPHERICAL AND HYPERBOLIC SPACES 99 5 GEO MAP UNI CATION 105 5 1 INTRODUCTION 105 5 2 HISTORICAL BACKGROUND 106 5 3 GEOMETRIC BACKGROUND 108 5 3 1 A NE SPACE 108 5 3 2 PROJECTIVE SPACE 109 5 4 THE UNI ED GEO MAP COMPUTATIONAL FRAMEWORK 110 5 4 1 GEO MAP UNI CATION 110 5 4 2 A SIMPLE EXAMPLE 112 5 4 3 EXPRESSING EUCLIDEAN OPERATIONS IN THE SURROUNDING GEOMETRIC ALGEBRA 113 5 5 APPLYING THE GEO MAP TECHNIQUE TO GEOMETRICAL OPTICS 114 5 5 1 SOME GEOMETRIC OPTICAL BACKGROUND 114 5 5 2 DETERMINING THE SECOND ORDER LAW OF RE ECTION FOR PLANAR LIGHT RAYS 115 5 5 3 INTERPRETING THE SECOND ORDER LAW OF RE ECTION GEOMETRICALLY 121 5 6 SUMMARY AND CONCLUSIONS 122 5 6 1 THE GEO MAP UNI CATION TECHNIQUE 122 5 6 2 ALGEBRAIC AND COMBINATORIAL CONSTRUCTION OF A GEOMETRIC ALGEBRA 123 5 7 ACKNOWLEDGEMENTS 123 5 8 APPENDIX CONSTRUCTION OF A GEOMETRIC ALGEBRA 123