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

Preface......Page 13
http://www.geometricalgebra.net......Page 16
1.1 AN EXAMPLE IN GEOMETRIC ALGEBRA......Page 18
1.2 HOW IT WORKS AND HOW IT’S DIFFERENT......Page 24
1.3 PROGRAMMING GEOMETRY......Page 32
1.4 THE STRUCTURE OF THIS BOOK......Page 34
1.4.2 PART II: MODELS OF GEOMETRY......Page 35
1.5 THE STRUCTURE OF THE CHAPTERS......Page 36
2 SPANNING ORIENTEDSUBSPACES......Page 37
2.1 VECTOR SPACES......Page 38
2.2 ORIENTED LINE ELEMENTS......Page 39
2.2.2 VISUALIZING VECTORS......Page 40
2.3 ORIENTED AREA ELEME......Page 41
2.3.2 INTRODUCING THE OUTER PRODUCT......Page 42
2.3.3 VISUALIZING BIVECTORS......Page 45
2.3.4 VISUALIZING BIVECTOR ADDITION......Page 46
2.4 ORIENTED VOLUME ELEMENTS......Page 47
2.4.2 ASSOCIATIVITY OF THE OUTER PRODUCT......Page 49
2.4.3 VISUALIZATION OF TRIVECTORS......Page 50
2.6 SCALARS INTERPRETED GEOMETRICALLY......Page 51
2.7.1 SOLVING LINEAR EQUATIONS......Page 53
2.7.2 INTERSECTING PLANAR LINES......Page 55
2.8 HOMOGENEOUS SUBSPACE REPRESENTATION......Page 56
2.8.3 NONMETRIC LENGTHS, AREAS, AND VOLUMES......Page 57
2.9.1 BLADES AND GRADES......Page 58
2.9.2 THE LADDER OF SUBSPACES......Page 59
2.9.3 k-BLADES VERSUS k-VECTORS......Page 60
2.9.4 THE GRASSMANN ALGEBRA OF MULTIVECTORS......Page 61
2.9.5 REVERSION AND GRADE INVOLUTION......Page 63
2.10 SUMMARY OF OUTER PRODUCT PROPERTIES......Page 64
2.11 FURTHER READING......Page 65
2.13.1 DRAWING BIVECTORS......Page 71
2.13 PROGRAMMING EXAMPLES AND EXERCISES......Page 67
3 METRIC PRODUCTSOF SUBSPACES......Page 79
3.1 SIZING UP SUBSPACES......Page 80
3.1.2 DEFINITION OF THE SCALAR PRODUCT......Page 81
3.1.4 THE ANGLE BETWEEN SUBSPACES......Page 82
3.2.1 IMPLICIT DEFINITION OF CONTRACTION ......Page 85
3.2.2 COMPUTING THE CONTRACTION EXPLICITLY......Page 87
3.2.3 ALGEBRAIC SUBTLETIES......Page 88
3.3 GEOMETRIC INTERPRETATION OF THECONTRACTION......Page 89
3.4 THE OTHER CONTRACTION......Page 91
3.5 ORTHOGONALITY AND DUALITY......Page 92
3.5.3 ORTHOGONAL COMPLEMENT AND DUALITY......Page 94
3.5.4 THE DUALITY RELATIONSHIPS......Page 96
3.6 ORTHOGONAL PROJECTION OF SUBSPACES......Page 97
3.7.1 USES OF THE CROSS PRODUCT......Page 100
3.7.2 THE CROSS PRODUCT INCORPORATED......Page 101
3.8 APPLICATION: RECIPROCAL FRAMES......Page 103
3.9 FURTHER READING......Page 105
3.11 PROGRAMMING EXAMPLES AND EXERCISES......Page 107
3.11.3 RECIPROCAL FRAMES......Page 109
4 LINEAR TRANSFORMATIONSOF SUBSPACES......Page 113
4.1 LINEAR TRANSFORMATIONS OF VECTORS......Page 114
4.2 OUTERMORPHISMS: LINEARTRANSFORMATIONS OF BLADES......Page 115
4.2.1 MOTIVATION OF THE OUTERMORPHISM......Page 116
4.2.2 EXAMPLES OF OUTERMORPHISMS......Page 117