Introduction to geometric algebra computing: Computing with Circles and Lines
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Milton
Chapman and Hall/CRC
[2018]
|
| Series: | Computer Vision Ser
|
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | Description based on publisher supplied metadata and other sources |
| Physical Description: | 1 Online-Ressource (212 Seiten) |
| ISBN: | 9781315152172 |
| DOI: | 10.1201/9781315152172 |
Staff View
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| contents | Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Table of Contents -- Foreword -- Preface -- Acknowledgments -- Chapter 1 ▪ Introduction -- 1.1 Geometric Algebra -- 1.2 Geometric Algebra Computing -- 1.3 Outline -- 1.3.1 SECTION I : Tutorial -- 1.3.2 SECTION II : Mathematical Foundations -- 1.3.3 SECTION III : Applications -- 1.3.4 SECTION IV : Geometric Algebra at School -- Section I: Tutorial -- Chapter 2 ▪ Compass Ruler Algebra in a Nutshell -- 2.1 Geometric Objects -- 2.2 Angles and Distances -- 2.3 Transformations -- Chapter 3 ▪ GAALOP Tutorial for Compass Ruler Algebra -- 3.1 GAALOP and GAALOPscript -- 3.2 Geometric Objects -- 3.2.1 Point -- 3.2.2 Circle -- 3.2.3 Line -- 3.2.4 Point pair -- 3.2.5 Perpendicular Bisector -- 3.2.6 The Difference of two Points -- 3.2.7 The Sum of Points -- 3.3 Angles and Distances -- 3.3.1 Distance Point-Line -- 3.3.2 Angle between two Lines -- 3.3.3 Distance between two Circles -- 3.4 Geometric Transformations -- 3.4.1 Reflections -- 3.4.1.1 Rotations based on reflections -- 3.4.1.2 Translations based on reflections -- 3.4.1.3 Inversions -- 3.4.2 Rotors -- 3.4.3 Translators -- 3.4.4 Motors -- Section II Mathematical Foundations -- Chapter 4 ▪ Mathematical Basics and 2D Euclidean Geometric Algebra -- 4.1 The Basic Algebraic Elements of Geometric Algebra -- 4.2 The Products of Geometric Algebra -- 4.2.1 The Outer Product -- 4.2.2 The Inner Product -- 4.2.3 The Geometric Product -- 4.3 The Imaginary Unit in Geometric Algebra -- 4.4 The Inverse -- 4.5 The Dual -- 4.6 The Reverse -- Chapter 5 ▪ Compass Ruler Algebra and Its Geometric Objects -- 5.1 The Algebraic Structure -- 5.2 The Basic Geometric Entities and Their Null Spaces -- 5.3 Points -- 5.4 Lines -- 5.5 Circles -- 5.6 Normalized Objects -- 5.7 The Difference of Two Points -- 5.8 The Sum of Points -- 5.9 The Meaning of E0 and E∞ 5.10 Line as a Limit of a Circle -- 5.11 Point Pairs -- Chapter 6 ▪ Intersections in Compass Ruler Algebra -- 6.1 The IPNS of the Outer Product of Two Vectors -- 6.2 The Role of E1 ˄ E2 -- 6.3 The Intersection of Two Lines -- 6.4 The Intersection of Two Parallel Lines -- 6.5 The Intersection of Circle-Line -- 6.6 Oriented Points -- 6.7 The Intersection of Circles -- Chapter 7 ▪ Distances and Angles in Compass Ruler Algebra -- 7.1 Distance between Points -- 7.2 Distance between a Point and a Line -- 7.3 Angles between Lines -- 7.4 Distance between a Line and a Circle -- 7.5 Distance Relations between a Point and a Circle -- 7.6 Is a Point Inside or Outside a Circle? -- 7.7 Distance to the Horizon -- 7.8 Distance Relations between Two Circles -- 7.8.1 Distance between Circles with Equal Radii -- 7.8.2 Example of Circles with Di erent Radii -- 7.8.3 General Solution -- 7.8.4 Geometric Meaning -- Chapter 8 ▪ Transformations of Objects in Compass Ruler Algebra -- 8.1 Reflection at the Coordinate Axes -- 8.2 The Role of E1 ˄ E2 -- 8.3 Arbitrary Reflections -- 8.4 Rotor Based on Reflections -- 8.5 Translation -- 8.6 Rigid Body Motion -- 8.7 Multivector Exponentials -- 8.8 Inversion and the Center of a Circle or Point Pair -- Section III Applications -- Chapter 9 ▪ Robot Kinematics Using GAALOP -- 9.1 Inverse Kinematics Using GAALOP -- 9.2 Steps to Reach the Target -- 9.3 Movement Toward the Target -- Chapter 10 ▪ Detection of Circles and Lines in Images Using GAALOP -- 10.1 CGAVS Algorithm -- 10.2 GAALOP Implementation -- Chapter 11 ▪ Visibility Application in 2D Using GAALOP -- 11.1 Is a Circle Outside a 2D Cone? -- 11.2 Visibility Sequence -- Chapter 12 ▪ Runtime-Performance Using GAALOP -- 12.1 C Code of the Standard CGAVS Implementation -- 12.2 Avoiding Normalizations -- 12.3 Avoiding Explicit Statement Computations -- 12.4 New CGAVS Algorithm 12.5 Hardware Implementation Based on GAALOP -- Chapter 13 ▪ Fitting of Lines or Circles into Sets of Points -- 13.1 Distance Measure -- 13.2 Least-Squares Approach -- Chapter 14 ▪ CRA-Based Robotic Snake Control -- 14.1 Robotic Snakes -- 14.2 Direct Kinematics -- 14.2.1 Singular positions -- 14.3 Differential Kinematics -- 14.4 3-Link Snake Model -- Chapter 15 ▪ Expansion to 3D Computations -- 15.1 CLUCalc for 3D Visualizations -- 15.2 The Geometric Objects of CGA -- 15.3 Angles and Distances in 3D -- 15.4 3D Transformations -- 15.5 CLUCalc Implementation of the Snake Robot Control -- 15.6 3D Computations with GAALOP -- 15.7 Visibility Application in 3D -- 15.8 Conclusion of the Engineering Part -- Section IV Geometric Algebra at School -- Chapter 16 ▪ Geometric Algebra for Mathematical Education -- 16.1 Basic DGS Functionality Based on GAALOP -- 16.2 Geometric Constructions Based on Compass Ruler Algebra -- 16.3 Deriving of Formulae -- 16.4 Proving Geometric Relationships -- 16.5 Outlook -- Chapter 17 ▪ Space-Time Algebra in School and Application -- 17.1 The Algebraic Structure of Space-Time Algebra -- 17.2 Space-Time Algebra at School -- 17.3 A Faraday Example for Mathematica's Opencllink -- Bibliography -- Index |
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| dewey-ones | 512 - Algebra |
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| discipline | Informatik Mathematik |
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| doi_str_mv | 10.1201/9781315152172 |
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| index_date | 2024-07-03T18:01:24Z |
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| isbn | 9781315152172 |
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| spelling | Hildenbrand, Dietmar 1961- Verfasser (DE-588)132458772 aut Introduction to geometric algebra computing Dietmar Hildenbrand Milton Chapman and Hall/CRC [2018] ©2018 1 Online-Ressource (212 Seiten) txt rdacontent c rdamedia cr rdacarrier Computer Vision Ser Description based on publisher supplied metadata and other sources 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 978-1-4987-4841-4 https://doi.org/10.1201/9781315152172 Verlag URL des Erstveröffentlichers Volltext |
| spellingShingle | Hildenbrand, Dietmar 1961- Introduction to geometric algebra computing Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Table of Contents -- Foreword -- Preface -- Acknowledgments -- Chapter 1 ▪ Introduction -- 1.1 Geometric Algebra -- 1.2 Geometric Algebra Computing -- 1.3 Outline -- 1.3.1 SECTION I : Tutorial -- 1.3.2 SECTION II : Mathematical Foundations -- 1.3.3 SECTION III : Applications -- 1.3.4 SECTION IV : Geometric Algebra at School -- Section I: Tutorial -- Chapter 2 ▪ Compass Ruler Algebra in a Nutshell -- 2.1 Geometric Objects -- 2.2 Angles and Distances -- 2.3 Transformations -- Chapter 3 ▪ GAALOP Tutorial for Compass Ruler Algebra -- 3.1 GAALOP and GAALOPscript -- 3.2 Geometric Objects -- 3.2.1 Point -- 3.2.2 Circle -- 3.2.3 Line -- 3.2.4 Point pair -- 3.2.5 Perpendicular Bisector -- 3.2.6 The Difference of two Points -- 3.2.7 The Sum of Points -- 3.3 Angles and Distances -- 3.3.1 Distance Point-Line -- 3.3.2 Angle between two Lines -- 3.3.3 Distance between two Circles -- 3.4 Geometric Transformations -- 3.4.1 Reflections -- 3.4.1.1 Rotations based on reflections -- 3.4.1.2 Translations based on reflections -- 3.4.1.3 Inversions -- 3.4.2 Rotors -- 3.4.3 Translators -- 3.4.4 Motors -- Section II Mathematical Foundations -- Chapter 4 ▪ Mathematical Basics and 2D Euclidean Geometric Algebra -- 4.1 The Basic Algebraic Elements of Geometric Algebra -- 4.2 The Products of Geometric Algebra -- 4.2.1 The Outer Product -- 4.2.2 The Inner Product -- 4.2.3 The Geometric Product -- 4.3 The Imaginary Unit in Geometric Algebra -- 4.4 The Inverse -- 4.5 The Dual -- 4.6 The Reverse -- Chapter 5 ▪ Compass Ruler Algebra and Its Geometric Objects -- 5.1 The Algebraic Structure -- 5.2 The Basic Geometric Entities and Their Null Spaces -- 5.3 Points -- 5.4 Lines -- 5.5 Circles -- 5.6 Normalized Objects -- 5.7 The Difference of Two Points -- 5.8 The Sum of Points -- 5.9 The Meaning of E0 and E∞ 5.10 Line as a Limit of a Circle -- 5.11 Point Pairs -- Chapter 6 ▪ Intersections in Compass Ruler Algebra -- 6.1 The IPNS of the Outer Product of Two Vectors -- 6.2 The Role of E1 ˄ E2 -- 6.3 The Intersection of Two Lines -- 6.4 The Intersection of Two Parallel Lines -- 6.5 The Intersection of Circle-Line -- 6.6 Oriented Points -- 6.7 The Intersection of Circles -- Chapter 7 ▪ Distances and Angles in Compass Ruler Algebra -- 7.1 Distance between Points -- 7.2 Distance between a Point and a Line -- 7.3 Angles between Lines -- 7.4 Distance between a Line and a Circle -- 7.5 Distance Relations between a Point and a Circle -- 7.6 Is a Point Inside or Outside a Circle? -- 7.7 Distance to the Horizon -- 7.8 Distance Relations between Two Circles -- 7.8.1 Distance between Circles with Equal Radii -- 7.8.2 Example of Circles with Di erent Radii -- 7.8.3 General Solution -- 7.8.4 Geometric Meaning -- Chapter 8 ▪ Transformations of Objects in Compass Ruler Algebra -- 8.1 Reflection at the Coordinate Axes -- 8.2 The Role of E1 ˄ E2 -- 8.3 Arbitrary Reflections -- 8.4 Rotor Based on Reflections -- 8.5 Translation -- 8.6 Rigid Body Motion -- 8.7 Multivector Exponentials -- 8.8 Inversion and the Center of a Circle or Point Pair -- Section III Applications -- Chapter 9 ▪ Robot Kinematics Using GAALOP -- 9.1 Inverse Kinematics Using GAALOP -- 9.2 Steps to Reach the Target -- 9.3 Movement Toward the Target -- Chapter 10 ▪ Detection of Circles and Lines in Images Using GAALOP -- 10.1 CGAVS Algorithm -- 10.2 GAALOP Implementation -- Chapter 11 ▪ Visibility Application in 2D Using GAALOP -- 11.1 Is a Circle Outside a 2D Cone? -- 11.2 Visibility Sequence -- Chapter 12 ▪ Runtime-Performance Using GAALOP -- 12.1 C Code of the Standard CGAVS Implementation -- 12.2 Avoiding Normalizations -- 12.3 Avoiding Explicit Statement Computations -- 12.4 New CGAVS Algorithm 12.5 Hardware Implementation Based on GAALOP -- Chapter 13 ▪ Fitting of Lines or Circles into Sets of Points -- 13.1 Distance Measure -- 13.2 Least-Squares Approach -- Chapter 14 ▪ CRA-Based Robotic Snake Control -- 14.1 Robotic Snakes -- 14.2 Direct Kinematics -- 14.2.1 Singular positions -- 14.3 Differential Kinematics -- 14.4 3-Link Snake Model -- Chapter 15 ▪ Expansion to 3D Computations -- 15.1 CLUCalc for 3D Visualizations -- 15.2 The Geometric Objects of CGA -- 15.3 Angles and Distances in 3D -- 15.4 3D Transformations -- 15.5 CLUCalc Implementation of the Snake Robot Control -- 15.6 3D Computations with GAALOP -- 15.7 Visibility Application in 3D -- 15.8 Conclusion of the Engineering Part -- Section IV Geometric Algebra at School -- Chapter 16 ▪ Geometric Algebra for Mathematical Education -- 16.1 Basic DGS Functionality Based on GAALOP -- 16.2 Geometric Constructions Based on Compass Ruler Algebra -- 16.3 Deriving of Formulae -- 16.4 Proving Geometric Relationships -- 16.5 Outlook -- Chapter 17 ▪ Space-Time Algebra in School and Application -- 17.1 The Algebraic Structure of Space-Time Algebra -- 17.2 Space-Time Algebra at School -- 17.3 A Faraday Example for Mathematica's Opencllink -- Bibliography -- Index 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 |
| title_auth | Introduction to geometric algebra computing |
| title_exact_search | Introduction to geometric algebra computing |
| title_exact_search_txtP | Introduction to Geometric Algebra Computing Computing with Circles and Lines |
| title_full | Introduction to geometric algebra computing Dietmar Hildenbrand |
| title_fullStr | Introduction to geometric algebra computing Dietmar Hildenbrand |
| title_full_unstemmed | Introduction to geometric algebra computing Dietmar Hildenbrand |
| title_short | Introduction to geometric algebra computing |
| title_sort | introduction to geometric algebra computing |
| 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 |
| url | https://doi.org/10.1201/9781315152172 |
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