Introduction to Geometric Algebra Computing; First Edition

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 Dierent 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