Introduction to geometric algebra computing Computing with Circles and Lines

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Bibliographische Detailangaben
1. Verfasser:	Hildenbrand, Dietmar (VerfasserIn)
Format:	Elektronisch E-Book
Sprache:	English
Veröffentlicht:	Milton Chapman and Hall/CRC [2018]
Schriftenreihe:	Computer Vision Ser
Schlagworte:	Geometrische Algebra Computeralgebra
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!

Inhaltsangabe:

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

Introduction to geometric algebra computing Computing with Circles and Lines

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