Foundations of geometric algebra computing:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2013
|
| Series: | Geometry and computing
8 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XXVII, 196 S. Ill., graph. Darst. |
| ISBN: | 9783642317934 3642317936 |
Staff View
MARC
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Record in the Search Index
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| adam_text |
IMAGE 1
CONTENTS
1 INTRODUCTION 1
1.1 THE BENEFITS OF GEOMETRIC ALGEBRA 2
1.2 THE BENEFITS OF GEOMETRIC ALGEBRA COMPUTING 5
1.3 HISTORY O F GEOMETRIC ALGEBRA COMPUTING 7
1.4 OVERVIEW 11
1.5 OUTLINE 12
1.5.1 P A R T I 12
1.5.2 PART II 13
1.5.3 PART III 13
PART I MATHEMATICAL FOUNDATIONS
2 MATHEMATICAL INTRODUCTION 17
2.1 THE BASIC ALGEBRAIC ELEMENTS O F GEOMETRIC ALGEBRA 17
2.2 THE PRODUCTS O F GEOMETRIC ALGEBRA 18
2.2.1 THE OUTER PRODUCT 18
2.2.2 THE INNER PRODUCT 20
2.2.3 THE GEOMETRIC PRODUCT 20
2.3 EUCLIDEAN GEOMETRIC ALGEBRA 23
2.4 PROJECTIVE GEOMETRIC ALGEBRA 25
3 CONFORMAL GEOMETRIC ALGEBRA 27
3.1 THE BASIC GEOMETRIC ENTITIES 28
3.1.1 POINTS 29
3.1.2 SPHERES 30
3.1.3 PLANES 30
3.1.4 CIRCLES 30
3.1.5 LINES 31
3.1.6 POINT PAIRS 31
3.2 IPNS AND OPNS 31
3.3 THE CENTER OF A SPHERE, CIRCLE, OR POINT PAIR 32
X V
HTTP://D-NB.INFO/1024100189
IMAGE 2
X V J C O N T E N T S
3.4 DISTANCES AND ANGLES 33
3.4.1 DISTANCES 34
3.4.2 ANGLES 38
3.5 TRANSFORMATIONS 39
3.5.1 ROTATION 39
3.5.2 TRANSLATION 4 0
3.6 RIGID-BODY MOTION 4 2
3.7 THE HORIZON EXAMPLE 4 2
4 MAPLE AND THE IDENTIFICATION O F QUATERNIONS AND OTHER ALGEBRAS 45 4.1
USING MAPLE FOR SYMBOLIC GEOMETRIC ALGEBRA COMPUTING 45
4.2 COMPLEX NUMBERS 47
4.3 QUATERNIONS 48
4.3.1 THE IMAGINARY UNITS 5 0
4.3.2 PURE QUATERNIONS AND THEIR GEOMETRIC PRODUCT 51
4.3.3 ROTATIONS BASED ON UNIT QUATERNIONS 52
4.4 PLUCKER COORDINATES 53
4.5 DUAL NUMBERS 55
4.6 DUAL QUATERNIONS 56
5 FITTING O F PLANES OR SPHERES TO SETS O F POINTS 61
5.1 THE ROLE O F INFINITY _ 61
5.1.1 SPHERE OF INFINITE RADIUS 62
5.1.2 POINT AT INFINITY 62
5.1.3 PLANE AT INFINITE DISTANCE FROM THE ORIGIN 63
5.1.4 PLANES AS A LIMIT O F A SPHERE 64
5.2 DISTANCE MEASURE 65
5.3 LEAST-SQUARES APPROACH 65
5.4 EXAMPLE 67
PART II INTERACTIVE AND VISUAL GEOMETRIC ALGEBRA COMPUTING
6 A TUTORIAL ON GEOMETRIC ALGEBRA USING CLUCALC 71
6.1 BLADES AND VECTORS 73
6.2 THE PRODUCTS OF GEOMETRIC ALGEBRA 74
6.2.1 THE OUTER PRODUCT AND PARALLELNESS 74
6.2.2 THE INNER PRODUCT AND PERPENDICULARITY 77
6.2.3 THE GEOMETRIC PRODUCT AND INVERTIBILITY 78
6.3 GEOMETRIC OPERATIONS 81
6.3.1 PROJECTION AND REJECTION 82
6.3.2 REFLECTION 83
6.3.3 ROTATION IN 2D 84
6.3.4 ROTATION IN 3D 86
6.4 CONFORMAL GEOMETRIC ALGEBRA 86
6.4.1 VECTORS IN CGA 87
6.4.2 BIVECTORS IN CGA 89
IMAGE 3
C O N T E N T S X V I I
6.4.3 DUAL VECTORS IN CGA 91
6.4.4 IS A POINT INSIDE OR OUTSIDE THE CIRCUMCIRCLE OF A TRIANGLE? 92
6.4.5 INTERSECTIONS 93
6.4.6 REFLECTION 96
6.4.7 PROJECTION 97
6.5 CLUCALC IMPLEMENTATION O F THE HORIZON EXAMPLE 98
6.6 CLUCALC IMPLEMENTATION O F MOTIONS 99
7 INVERSE KINEMATICS O F A SIMPLE ROBOT 101
7.1 COMPUTATION O F PO 102
7.2 COMPUTATION O F P2 103
7.3 COMPUTATION O F P\ 104
7.4 COMPUTATION O F THE JOINT ANGLES 105
8 ROBOT GRASPING AN OBJECT 107
8.1 THE GEOMETRIC ALGEBRA ALGORITHM 107
8.1.1 COMPUTATION O F THE BOUNDING VOLUME OF THE OBJECT 107 8.1.2
COMPUTATION O F THE GRASPING CIRCLE Z, 108
8.1.3 GRIPPER CIRCLE 109
8.1.4 ESTIMATION O F TRANSLATION AND ROTATION 110
8.2 THE ALGORITHM USING CLUCALC I L L
8.3 GEOMETRIC ALGEBRA VERSUS CONVENTIONAL MATHEMATICS 115
8.3.1 THE BASE CIRCLE AND ITS CENTER 115
8.3.2 THE TRANSFORMATION O F THE GRIPPER 116
PART III RUNTIME PERFORMANCE O F GEOMETRIC ALGEBRA COMPUTING
9 EFFICIENT COMPUTER ANIMATION APPLICATION IN CGA 121
9.1 OPTIMIZATIONS BASED ON QUATERNIONS 121
9.1.1 DIRECT COMPUTATION O F QUATERNIONS 122
9.1.2 EFFICIENT COMPUTATION O F QUATERNIONS 123
9.2 THE INVERSE KINEMATICS ALGORITHM 123
9.2.1 COMPUTATION O F THE SWIVEL PLANE 124
9.2.2 THE ELBOW POINT P E 125
9.2.3 CALCULATION O F THE ELBOW QUATERNION Q E 126
9.2.4 ROTATION TO THE ELBOW POSITION 127
9.2.5 ROTATION TO THE WRIST LOCATION 128
9.3 APPROACHES TO RUNTIME OPTIMIZATION 129
9.3.1 OPTIMIZATION WITH GAIGEN 2 129
9.3.2 OPTIMIZATION WITH MAPLE 134
9.4 RESULTS 138
IMAGE 4
XVIII C O N T E N T S
10 USING GAALOP FOR HIGH-PERFORMANCE GEOMETRIC
ALGEBRA COMPUTING 141
10.1 THE HORIZON EXAMPLE WITH GAALOP 141
10.2 THE GEOMETRIC ALGEBRA COMPUTING APPROACH 142
10.3 TABLE-BASED COMPILATION APPROACH 145
10.3.1 MULTIPLICATION TABLES 146
10.3.2 TABLE-BASED MULTIPLICATION ALGORITHM 148
10.3.3 EXAMPLE 148
10.3.4 CASCADING MULTIPLICATIONS 151
10.3.5 LINEAR OPERATION TABLES 153
10.3.6 MULTIPLICATION TABLES WITH A NON-EUCLIDEAN METRIC 153 10.3.7
ADDITIONAL SYMBOLIC OPTIMIZATIONS USING MAXIMA 154
11 COLLISION DETECTION USING THE GAALOP PRECOMPILER 155
11.1 BASIC CONCEPT O F GAALOP GPC 155
11.2 THE HORIZON EXAMPLE REVISITED IN GAALOP GPC FOR C++ 156
11.3 COLLISION DETECTION 158
12 THE GAALOP PRECOMPILER FOR GPUS 161
12.1 STRIDED ARRAYS 161
12.2 THE HORIZON EXAMPLE ON A GPU 162
12.2.1 OPENCL IMPLEMENTATION 162
12.2.2 CUDA IMPLEMENTATION 163
12.3 LIST O F MULTIVECTOR FUNCTIONS 164
13 MOLECULAR DYNAMICS USING GAALOP GPC FOR OPENCL 165
13.1 MOLECULAR DYNAMICS IN A NUTSHELL 165
13.2 SOFTWARE ARCHITECTURE 167
13.3 INITIALIZATION 168
13.4 VELOCITY VERLET INTEGRATION STEP 1 170
13.5 ACCUMULATION OF FORCES PER ATOM 171
13.6 VELOCITY VERLET INTEGRATION STEP 2 176
14 GEOMETRIC ALGEBRA COMPUTERS 1 79
14.1 FPGA IMPLEMENTATION OF GEOMETRIC ALGEBRA ALGORITHMS 1 179
14.2 ADAPTATION O F GEOMETRIC ALGEBRA TO CURRENT COMPUTER ARCHITECTURES
181
14.3 GEOMETRIC ALGEBRA PARALLELISM PROGRAMS (GAPP) 182
14.3.1 EXAMPLE 184
14.3.2 PARALLELIZATION CONCEPTS SUPPORTED BY GAPP 187
14.4 GEOMETRIC ALGEBRA COMPUTERS BASED ON GAALOP 188
REFERENCES 189
INDEX 195 |
| any_adam_object | 1 |
| author | Hildenbrand, Dietmar 1961- |
| author_GND | (DE-588)132458772 |
| author_facet | Hildenbrand, Dietmar 1961- |
| author_role | aut |
| author_sort | Hildenbrand, Dietmar 1961- |
| author_variant | d h dh |
| building | Verbundindex |
| bvnumber | BV041109678 |
| classification_rvk | SK 240 ST 230 ST 600 |
| ctrlnum | (OCoLC)827072012 (DE-599)DNB1024100189 |
| dewey-full | 512.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.5 |
| dewey-search | 512.5 |
| dewey-sort | 3512.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| format | Book |
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| indexdate | 2024-08-03T00:45:30Z |
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| isbn | 9783642317934 3642317936 |
| language | English |
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| physical | XXVII, 196 S. Ill., graph. Darst. |
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| series | Geometry and computing |
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| spelling | Hildenbrand, Dietmar 1961- Verfasser (DE-588)132458772 aut Foundations of geometric algebra computing Dietmar Hildenbrand Berlin [u.a.] Springer 2013 XXVII, 196 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Geometry and computing 8 Literaturangaben Geometrische Algebra (DE-588)4156707-9 gnd rswk-swf Computeralgebra (DE-588)4010449-7 gnd rswk-swf Geometrische Algebra (DE-588)4156707-9 s Computeralgebra (DE-588)4010449-7 s DE-604 Erscheint auch als Online-Ausgabe Foundations of Geometric Algebra Computing Geometry and computing 8 (DE-604)BV022959018 8 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4072875&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026085909&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hildenbrand, Dietmar 1961- Foundations of geometric algebra computing Geometry and computing Geometrische Algebra (DE-588)4156707-9 gnd Computeralgebra (DE-588)4010449-7 gnd |
| subject_GND | (DE-588)4156707-9 (DE-588)4010449-7 |
| title | Foundations of geometric algebra computing |
| title_auth | Foundations of geometric algebra computing |
| title_exact_search | Foundations of geometric algebra computing |
| title_full | Foundations of geometric algebra computing Dietmar Hildenbrand |
| title_fullStr | Foundations of geometric algebra computing Dietmar Hildenbrand |
| title_full_unstemmed | Foundations of geometric algebra computing Dietmar Hildenbrand |
| title_short | Foundations of geometric algebra computing |
| title_sort | foundations of geometric algebra computing |
| topic | Geometrische Algebra (DE-588)4156707-9 gnd Computeralgebra (DE-588)4010449-7 gnd |
| topic_facet | Geometrische Algebra Computeralgebra |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4072875&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026085909&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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