Introduction to geometric algebra computing: computing with circles and lines
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton, FL ; London ; New York, NY
CRC Press
[2019]
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xviii, 193 Seiten Illustrationen |
| ISBN: | 9781498748384 1498748384 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804178180305584128 |
|---|---|
| adam_text | Contents
Foreword xiii
Preface xv
Acknowledgments xvii
Chapter 1 ■ Introduction 1
1.1 GEOMETRIC ALGEBRA 1
1.2 GEOMETRIC ALGEBRA COMPUTING 2
1.3 OUTLINE 3
1.3.1 SECTION I : Tutorial 3
1.3.2 SECTION II : Mathematical Foundations 4
1.3.3 SECTION III : Applications 4
1.3.4 SECTION IV : Geometric Algebra at School 5
Section I Tutorial
Chapter 2 ■ Compass Ruler Algebra in a Nutshell 9
2.1 GEOMETRIC OBJECTS 10
2.2 ANGLES AND DISTANCES 11
2.3 TRANSFORMATIONS 12
Chapter 3 ■ GAALQP Tutorial for Compass Ruler Algebra 13
3.1 GAALOP AND GAALOPSCRIPT 14
3.2 GEOMETRIC OBJECTS 16
3.2.1 Point 17
3.2.2 Circle 18
• •
Vll
viii ■ Contents
3.2.3 Line 20
3.2.4 Point pair 21
3.2.5 Perpendicular Bisector 22
3.2.6 The Difference of two Points 23
3.2.7 The Sum of Points 24
3.3 ANGLES AND DISTANCES 27
3.3.1 Distance Point-Line 27
3.3.2 Angle between two Lines 28
3.3.3 Distance between two Circles 31
3.4 GEOMETRIC TRANSFORMATIONS 32
3.4.1 Reflections 32
3.4.1.1 Rotations based on reflections 35
3.4.1.2 Translations based on reflections 36
3.4.1.3 Inversions 36
3.4.2 Rotors 39
3.4.3 Translators 40
3.4.4 Motors 41
Section II Mathematical Foundations
Chapter 4 ■ Mathematical Basics and 2D Euclidean
_______________Geometric Algebra______________________________45
4.1 THE BASIC ALGEBRAIC ELEMENTS OF GEOMETRIC
ALGEBRA 45
4.2 THE PRODUCTS OF GEOMETRIC ALGEBRA 46
4.2.1 The Outer Product 47
4.2.2 The Inner Product 48
4.2.3 The Geometric Product 49
4.3 THE IMAGINARY UNIT IN GEOMETRIC ALGEBRA 49
4.4 THE INVERSE 50
4.5 THE DUAL 51
4.6 THE REVERSE 51
Contents ■ ix
Chapter 5 ■ Compass Ruler Algebra and Its Geometric
_________Objects 53
5.1 THE ALGEBRAIC STRUCTURE 54
5.2 THE BASIC GEOMETRIC ENTITIES AND THEIR NULL
SPACES 55
5.3 POINTS 56
5.4 LINES 57
5.5 CIRCLES 58
5.6 NORMALIZED OBJECTS 59
5.7 THE DIFFERENCE OF TWO POINTS 61
5.8 THE SUM OF POINTS 61
5.9 THE MEANING OF E0 AND 61
5.10 LINE AS A LIMIT OF A CIRCLE 62
5.11 POINT PAIRS 64
Chapter 6 ■ Intersections in Compass Ruler Algebra 67
6.1 THE IPNS OF THE OUTER PRODUCT OF TWO VECTORS 67
6.2 THE ROLE OF E1 A E2 68
6.3 THE INTERSECTION OF TWO LINES 68
6.4 THE INTERSECTION OF TWO PARALLEL LINES 69
6.5 THE INTERSECTION OF CIRCLE-LINE 70
6.6 ORIENTED POINTS 71
6.7 THE INTERSECTION OF CIRCLES 71
Chapter 7 ■ Distances and Angles in Compass Ruler Algebra 73
7.1 DISTANCE BETWEEN POINTS 74
7.2 DISTANCE BETWEEN A POINT AND A LINE 75
7.3 ANGLES BETWEEN LINES 76
7.4 DISTANCE BETWEEN A LINE AND A CIRCLE 77
7.5 DISTANCE RELATIONS BETWEEN A POINT AND A CIRCLE 78
7.6 IS A POINT INSIDE OR OUTSIDE A CIRCLE? 79
7.7 DISTANCE TO THE HORIZON 81
7.8 DISTANCE RELATIONS BETWEEN TWO CIRCLES 83
7.8.1 Distance between Circles with Equal Radii 83
x ■ Contents
7.8.2 Example of Circles with Different Radii 87
7.8.3 General Solution 91
7.8.4 Geometric Meaning 93
Chapter 8 ■ Transformations of Objects in Compass Ruler
Algebra 95
8.1 REFLECTION AT THE COORDINATE AXES 96
8.2 THE ROLE OF E A E? 97
8.3 ARBITRARY REFLECTIONS 98
8.4 ROTOR BASED ON REFLECTIONS 99
8.5 TRANSLATION 100
8.6 RIGID BODY MOTION 101
8.7 MULTIVECTOR EXPONENTIALS 102
8.8 INVERSION AND THE CENTER OF A CIRCLE OR POINT
PAIR 103
Section III Applications
Chapter 9 ■ Robot Kinematics Using GAALOP____________107
9.1 INVERSE KINEMATICS USING GAALOP 108
9.2 STEPS TO REACH THE TARGET 110
9.3 MOVEMENT TOWARD THE TARGET 112
Chapter 10» Detection of Circles and Lines in Images
_____________Using GAALOP__________________________117
10.1 CGAVS ALGORITHM 117
10.2 GAALOP IMPLEMENTATION 118
Chapter 11 ■ Visibility Application in 2D Using GAALOP 123
11.1 IS A CIRCLE OUTSIDE A 2D CONE?
11.2 VISIBILITY SEQUENCE
124
125
Contents ■ xi
Chapter 12 ■ Runtime-Performance Using GAALOP 127
12.1 C CODE OF THE STANDARD CGAVS IMPLEMENTATION 127
12.2 AVOIDING NORMALIZATIONS 129
12.3 AVOIDING EXPLICIT STATEMENT COMPUTATIONS 132
12.4 NEW CGAVS ALGORITHM 134
12.5 HARDWARE IMPLEMENTATION BASED ON GAALOP 135
Chapter 13« Fitting of Lines or Circles into Sets of Points 137
13.1 DISTANCE MEASURE 138
13.2 LEAST-SQUARES APPROACH 138
Chapter 14» CRA-Based Robotic Snake Control 141
14.1 ROBOTIC SNAKES 142
14.2 DIRECT KINEMATICS .. 143
14.2.1 Singular positions 145
14.3 DIFFERENTIAL KINEMATICS 145
14.4 3-LINK SNAKE MODEL 148
Chapter 15 ■ Expansion to 3D Computations_______________ 155
15.1 CLUCALC FOR 3D VISUALIZATIONS 156
15.2 THE GEOMETRIC OBJECTS OF CGA 157
15.3 ANGLES AND DISTANCES IN 3D 159
15.4 3D TRANSFORMATIONS 159
15.5 CLUCALC IMPLEMENTATION OF THE SNAKE ROBOT
CONTROL 161
15.6 3D COMPUTATIONS WITH GAALOP 163
15.7 VISIBILITY APPLICATION IN 3D 164
15.8 CONCLUSION OF THE ENGINEERING PART 165
Section IV Geometric Algebra at School
Chapter 16« Geometric Algebra for Mathematical Education 169
16.1 BASIC DGS FUNCTIONALITY BASED ON GAALOP 1 70
xii ■ Contents
16.2 GEOMETRIC CONSTRUCTIONS BASED ON COMPASS
RULER ALGEBRA 171
16.3 DERIVING OF FORMULAE 172
16.4 PROVING GEOMETRIC RELATIONSHIPS 174
16.5 OUTLOOK 175
Chapter 17« Space-Time Algebra in School and Application 1 77
1 7.1 THE ALGEBRAIC STRUCTURE OF SPACE-TIME ALGEBRA 1 77
17.2 SPACE-TIME ALGEBRA AT SCHOOL 178
1 7.3 A FARADAY EXAMPLE FOR MATHEMATICAL
OPENCLLINK 181
Index
193
|
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| classification_rvk | ST 600 |
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| discipline | Informatik |
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| isbn | 9781498748384 1498748384 |
| language | English |
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| physical | xviii, 193 Seiten Illustrationen |
| publishDate | 2019 |
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| spelling | Hildenbrand, Dietmar 1961- Verfasser (DE-588)132458772 aut Introduction to geometric algebra computing computing with circles and lines Boca Raton, FL ; London ; New York, NY CRC Press [2019] xviii, 193 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier 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 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030101928&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hildenbrand, Dietmar 1961- Introduction to geometric algebra computing computing with circles and lines Geometrische Algebra (DE-588)4156707-9 gnd Computeralgebra (DE-588)4010449-7 gnd |
| subject_GND | (DE-588)4156707-9 (DE-588)4010449-7 |
| title | Introduction to geometric algebra computing computing with circles and lines |
| title_auth | Introduction to geometric algebra computing computing with circles and lines |
| title_exact_search | Introduction to geometric algebra computing computing with circles and lines |
| title_full | Introduction to geometric algebra computing computing with circles and lines |
| title_fullStr | Introduction to geometric algebra computing computing with circles and lines |
| title_full_unstemmed | Introduction to geometric algebra computing computing with circles and lines |
| title_short | Introduction to geometric algebra computing |
| title_sort | introduction to geometric algebra computing computing with circles and lines |
| title_sub | computing with circles and lines |
| topic | Geometrische Algebra (DE-588)4156707-9 gnd Computeralgebra (DE-588)4010449-7 gnd |
| topic_facet | Geometrische Algebra Computeralgebra |
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