D. Klawitter: Clifford Algebras Geometric Modelling and Chain Geometries with Application in Kinematics


D. Klawitter, Clifford Algebras – Geometric Modelling and Chain Geometries with Application in Kinematics, Springer Spektrum, Springer, Berlin, 2015. Softcover ISBN 978-3-658-07617-7, Eb00k: ISBN 978-3-658-07618-4.

Summary: After revising known representations of the group of Euclidean displacements Daniel Klawitter gives a comprehensive introduction into Clifford algebras. The Clifford algebra calculus is used to construct new models that allow descriptions of the group of projective transformations and inversions with respect to hyperquadrics. Afterwards, chain geometries over Clifford algebras and their subchain geometries are examined. The author applies this theory and the developed methods to the homogeneous Clifford algebra model corresponding to Euclidean geometry. Moreover, kinematic mappings for special Cayley-Klein geometries are developed. These mappings allow a description of existing kinematic mappings in a unifying framework.

Contents (http://www.springer.com/cda/content/document/cda_downloaddocument/9783658076177-t1.pdf?SGWID=0-0-45-1482992-p177038851)

Foreword VII

Preface XI

Introduction 1

1 Models and Representations 5

1.1 Description of Displacements . . . . . . . . . . . . . . . 5

1.1.1 Homogeneous Matrices . . . . . . . . . . . . . . 5

1.1.2 Dual Quaternions . . . . . . . . . . . . . . . . . 6

1.1.3 Dual Orthogonal Matrices . . . . . . . . . . . . 13

1.2 Point Models for Lines and Displacements . . . . . . . 15

1.2.1 Klein’s Quadric . . . . . . . . . . . . . . . . . . . 15

1.2.2 Study’s Quadric . . . . . . . . . . . . . . . . . . 21

1.2.3 Study’s Sphere . . . . . . . . . . . . . . . . . . . 25

1.3 Geometric Algebras, Clifford Algebras . . . . . . . . . . 26

1.3.1 De.nition of a Geometric Algebra . . . . . . . . 26

1.3.2 Properties of Clifford Algebras . . . . . . . . . . 28

1.3.3 Pin and Spin Groups . . . . . . . . . . . . . . . 33

1.3.4 Matrix Representation of Clifford Algebras . . 36

1.3.5 Linear Transformation of the Vector Space . . 37

1.4 The HomogeneousModel . . . . . . . . . . . . . . . . . 38

1.4.1 The Construction . . . . . . . . . . . . . . . . . 38

1.4.2 Exterior Algebra . . . . . . . . . . . . . . . . . . 39

1.4.3 Homogeneous Model via projective Grassmann

Algebra . . . . . . . . . . . . . . . . . . . . . . . 41

1.5 The ConformalModel . . . . . . . . . . . . . . . . . . . 42

1.5.1 Construction of Conformal Geometric Algebra 43

1.5.2 Blades in CGA . . . . . . . . . . . . . . . . . . . 45

1.5.3 Conformal Transformations . . . . . . . . . . . . 48

XVI Contents

1.6 A Clifford algebraic Approach to Line Geometry . . . 50

1.6.1 Collineations and Correlations in the Image Space 51

1.6.2 Algebra Representation of Linear Line Manifolds 53

1.6.3 Transformations . . . . . . . . . . . . . . . . . . 55

1.6.4 Collineations as Spin Group . . . . . . . . . . . 61

1.6.5 Correlations as Pin Group . . . . . . . . . . . . 67

1.6.6 Singular projective Transformations . . . . . . . 72

1.7 A Clifford algebraic Approach to Lie Sphere Geometry 72

1.7.1 Lie’s Quadric . . . . . . . . . . . . . . . . . . . . 73

1.7.2 The homogeneous Clifford Algebra Model corresponding

to Lie Sphere Geometry . . . . . . . 74

1.8 A Clifford algebraic Approach to Study’s Quadric . . . 76

1.9 A Clifford algebraic Approach to Study’s Sphere . . . 77

1.10 Quadric Geometric Algebra . . . . . . . . . . . . . . . . 78

1.10.1 The Embedding . . . . . . . . . . . . . . . . . . 78

1.10.2 Geometric Entities . . . . . . . . . . . . . . . . . 80

1.10.3 Transformations . . . . . . . . . . . . . . . . . . 84

1.10.4 E.ect on Lines and Points . . . . . . . . . . . . 87

1.10.5 Subgroups . . . . . . . . . . . . . . . . . . . . . . 89

1.10.6 Generalization to higher Dimensions . . . . . . 94

2 Chain Geometry over Clifford Algebras 101

2.1 Chain Geometry . . . . . . . . . . . . . . . . . . . . . . 101

2.1.1 Distance Spaces . . . . . . . . . . . . . . . . . . 101

2.1.2 The projective Line over an L-algebra . . . . . 102

2.1.3 The Projective Linear Group PGL(R, 2) . . . . 104

2.1.4 The projective Line over a Subring . . . . . . . 105

2.2 Chain Geometry as Incidence Geometry . . . . . . . . 106

2.2.1 De.nition of a Cross Ratio . . . . . . . . . . . . 109

2.3 Quadric Chain Spaces . . . . . . . . . . . . . . . . . . . 114

2.4 Real Benz Planes . . . . . . . . . . . . . . . . . . . . . . 116

2.5 Jordan-Systems . . . . . . . . . . . . . . . . . . . . . . . 118

2.6 Contact Spaces . . . . . . . . . . . . . . . . . . . . . . . 121

2.7 Chain Geometries over Clifford Algebras . . . . . . . . 123

2.7.1 Grade-1 Subspace . . . . . . . . . . . . . . . . . 124

2.7.2 Pin and Spin Groups . . . . . . . . . . . . . . . 126

2.8 Quadric Chain Geometry . . . . . . . . . . . . . . . . . 131

2.8.1 Klein’s Quadric . . . . . . . . . . . . . . . . . . . 131

Contents XVII

2.8.2 Lie’s Quadric . . . . . . . . . . . . . . . . . . . . 133

2.8.3 Study’s Quadric . . . . . . . . . . . . . . . . . . 135

2.8.4 Study’s Sphere . . . . . . . . . . . . . . . . . . . 136

2.9 Quadric Chain Spaces for certain Spin Groups . . . . . 139

2.9.1 A Quadric Model for dual unit Quaternions . . 139

2.9.2 Other possible Quadric Models . . . . . . . . . 141

2.10 Cross Ratio of dual unit Quaternions . . . . . . . . . . 145

2.10.1 Subspaces on Study’s Quadric and Sub Chain

Geometries . . . . . . . . . . . . . . . . . . . . . 150

2.10.2 Application to line-symmetric Displacements . 151

2.10.3 Dual Quaternion Cross Ratio and Conics on

Study’s Quadric . . . . . . . . . . . . . . . . . . 156

2.11 Chains of Geometric Entities . . . . . . . . . . . . . . . 161

2.11.1 Quadric Geometric Algebra . . . . . . . . . . . 161

2.12 Biarc Construction . . . . . . . . . . . . . . . . . . . . . 165

2.12.1 Biarcs as touching Chains . . . . . . . . . . . . 168

2.12.2 Biarcs on Quadrics in three-dimensional Space 169

2.12.3 Klein’s Quadric . . . . . . . . . . . . . . . . . . . 176

2.12.4 Biarcs on the Dual Sphere . . . . . . . . . . . . 177

3 Kinematic Mappings for Spin Groups 181

3.1 Cayley-Klein Geometries and the homogeneous Model 181

3.1.1 Cayley-Klein Spaces . . . . . . . . . . . . . . . . 182

3.1.2 A homogeneous Model for Euclidean Geometry 184

3.2 KinematicMappings . . . . . . . . . . . . . . . . . . . . 185

3.2.1 Study’s kinematicMapping . . . . . . . . . . . . 185

3.2.2 A Mapping for planar Displacements . . . . . . 185

3.3 Kinematic Mappings via Clifford Algebras . . . . . . . 187

3.3.1 Study’s Mapping via Clifford Algebra . . . . . . 187

3.3.2 Blaschke’s and Gr¨unwald’s Mapping via Clifford

Algebra . . . . . . . . . . . . . . . . . . . . 190

3.4 Kinematic mappings for other Cayley-Klein Spaces . . 192

3.4.1 Two-dimensional Cayley-Klein Spaces . . . . . 192

3.4.2 Three-dimensional Cayley-Klein Spaces . . . . 193

3.4.3 Higher dimensional kinematic Mappings . . . . 195

3.5 Projective Varieties via kinematic Algebra Elements . 197

Conclusion 201

XVIII Contents

Index 203

List of Symbols 205

Acknowledgment 209

Bibliography 211

http://www.springer.com/978-3-658-07617-7

 

Source: http://www.springer.com/jp/book/9783658076177#aboutBook

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