Guide to Geometric Algebra in Practice

Guide to Geometric Algebra in Practice
	How to Read This Guide to Geometric Algebra in Practice
		Part I: Rigid Body Motion
		Part II: Interpolation and Tracking
		Part III: Image Processing
		Part IV: Theorem Proving and Combinatorics
		Part V: Applications of Line Geometry
		Part VI: Alternatives to Conformal Geometric Algebra
		Part VII: Towards Coordinate-Free Differential Geometry
			Part VIII: Tutorial Appendix
	Contents
	Contributors
Part I: Rigid Body Motion
	Chapter 1: Rigid Body Dynamics and Conformal Geometric Algebra
		1.1 Introduction
		1.2 Rigid Body Dynamics in 3D
			1.2.1 Kinematics
			1.2.2 Dynamics
		1.3 Rigid Body Dynamics in 5D CGA
			1.3.1 Introduction
			1.3.2 Setting up the Lagrangian
			1.3.3 The Equations of Motion and Conservation Laws
			1.3.4 Solving the Equations
				1.3.4.1 Counting Arguments
			1.3.5 An Example-The Dumbbell
			1.3.6 Including Moments and Forces
			1.3.7 Adding in Gravity
			1.3.8 The Angular Velocity Bivector Update Equation
			1.3.9 Collisions-One Body
			1.3.10 Collisions-Two Bodies
		1.4 Implementation
			1.4.1 Overview
			1.4.2 Update Equations
				1.4.2.1 Velocity Update
			1.4.3 Computer-Level Object Representation
			1.4.4 Results
		1.5 Exercises
		References
	Chapter 2: Estimating Motors from a Variety  of Geometric Data in 3D Conformal  Geometric Algebra
		2.1 Introduction
		2.2 The Linear Spaces M, B, and S
		2.3 Geometry of the Motors
		2.4 Estimating Motors
			2.4.1 Similarity Measures in CGA
				Points and Spheres
				Flats
				Directions
				Tangents
				Rounds
			2.4.2 Motor Estimation Problem Formulation
			2.4.3 Optimal Rotator and Translator Estimation
			2.4.4 Optimal Motor Estimation as an Eigenrotator Problem
		2.5 Examples
		2.6 Discussion
		2.7 Exercises
		References
	Chapter 3: Inverse Kinematics Solutions Using  Conformal Geometric Algebra
		3.1 Introduction
		3.2 Background
		3.3 FABRIK: An Iterative Inverse Kinematics Solver
		3.4 Using FABRIK for Hand Pose Tracking
			3.4.1 The Hand Geometry
				Calculating the Palm Joints
				Calculating the Finger Joints
			3.4.2 Trigonometric Solutions
				3.4.2.1 Nearest Point on a Sphere from a Point in Space
				3.4.2.2 Nearest Point on a Circle from a Point in Space
		3.5 Experimental Results
		3.6 Conclusions and Future Work
		3.7 Exercises
		References
	Chapter 4: Reconstructing Rotations and Rigid  Body Motions from Exact Point  Correspondences Through Reflections
		4.1 Introduction
		4.2 Method for Reconstructing Rigid Body Transformations from Point Correspondences Through Plane Reflections
		4.3 Formalization Using Geometric Algebra
			4.3.1 Proof of Correctness
			4.3.2 Reconstructing 3D Rigid Body Motions from Three Point Correspondences
		4.4 Reconstructing a Quaternion from Two Vector Correspondences
		4.5 Benchmarks
			4.5.1 Performance for 3D Rotations
			4.5.2 Performance for 3D Rigid Body Motions
		4.6 Discussion
			4.6.1 Null Reflectors
			4.6.2 The Scaling of Vectors
			4.6.3 The Determinant of the Reconstructed Versor
		4.7 Conclusion
		4.8 Exercises
		References
Part II: Interpolation and Tracking
	Chapter 5: Square Root and Logarithm of Rotors  in 3D Conformal Geometric Algebra  Using Polar Decomposition
		5.1 Rotor Interpolation for Conformal Motions
		5.2 Rotor Roots Through Polar Decomposition
			5.2.1 Sketch: How to Take the Square Root of a Rotor
			5.2.2 The Square Root of Rotors in 3D CGA
			5.2.3 Polar Decomposition of Invertible Even Elements in 3D CGA
			5.2.4 Determining the Square Root of a Symmetric Element
			5.2.5 The Polar Decomposition of Invertible Elements of R4,1+
			5.2.6 The Square Root of a Rotor
			5.2.7 Roots When 1+R Is Noninvertible
			5.2.8 Interpolation
			5.2.9 Special Cases of Rotor Roots
		5.3 Logarithms of Rotors in 3D CGA
			5.3.1 Logarithm of a Simple Rotor
			5.3.2 The Split Structure of a Rotor
			5.3.3 Exterior Derivative of a Rotor Transformation in 3D CGA
			5.3.4 Split of a 3D CGA Bivector into Commuting 2-Blades
			5.3.5 The Principal Logarithm of a 3D CGA Rotor
		5.4 Geometrical Interpretation of the Logarithm
			5.4.1 The 2-Blade Generators in the Bivector Split
			5.4.2 Orbits of Rotors
		5.5 Special Cases of the Rotor Logarithm
			5.5.1 Log(TS)
			5.5.2 Log(RST): Generalized Chasles Theorem for Euclidean Similarities
			5.5.3 Log(TV)
		5.6 Exercises
		References
	Chapter 6: Attitude and Position Tracking
		6.1 Kinematics in Geometric Algebra
		6.2 Attitude Computation and Kinematics in 3D
			6.2.1 Rotation Bivectors
				Finding an Expression for alpha
				Finding an Expression for B
				Completing the Derivation
		6.3 Practical Kinematics and Coning Motion
			6.3.1 Integration Schemes
			6.3.2 Comparative Simulations
		6.4 General Kinematics: Combining Rotation and Translation with CGA
			6.4.1 Generalised Velocities
			6.4.2 A Conformal Kinematic Equation for Bivectors
				Finding Expressions for alpha and B
				Finding Expressions for t
				Completing the Derivation
			6.4.3 Simulation Results
		6.5 Conclusion
		6.6 Exercises
		References
	Chapter 7: Calibration of Target Positions Using Conformal Geometric Algebra
		7.1 Introduction
		7.2 Problem Statement
		7.3 Solution Using Geometric Algebra
			7.3.1 Initial Estimate of Poses
			7.3.2 Accurate Estimate of Poses
				7.3.2.1 Constraints
					Distance Between Two Targets
					Distance Between Two Poses
				7.3.2.2 Parameterisation
			7.3.3 Reconstructing the Target Positions
		7.4 Results
		7.5 Conclusions
		7.6 Exercises
		References
Part III: Image Processing
	Chapter 8: Quaternion Atomic Function for Image Processing
		8.1 Introduction
		8.2 Atomic Functions
			8.2.1 The Atomic Function up(x)
			8.2.2 The Differentiator Atomic Function dup(x)
		8.3 Quaternion Algebra
		8.4 Quaternion Atomic Function Qup(x)
		8.5 Monogenic Signal and the Atomic Function
		8.6 Quaternion Wavelet Atomic Function Transform
		8.7 Radon Transform of Functionals
		8.8 Applications of the Quaternion Atomic Function Qup(x)
			8.8.1 Convolution with an Image
			8.8.2 Multi-resolution Analysis Using the Quaternion Wavelet Atomic Function
			8.8.3 Radon Transform for Circle Detection in Color Images Using the Quaternion Atomic Functions
		8.9 Conclusion
		8.10 Exercises
		References
	Chapter 9: Color Object Recognition Based  on a Clifford Fourier Transform
		9.1 Introduction
		9.2 A Clifford Fourier Transform for Color Image Processing
			9.2.1 The Shift Theorem for the Clifford Fourier Transform
			9.2.2 Computation of the Clifford Fourier Transform
		9.3 Generalized Color Fourier Descriptors
			9.3.1 Generalized Fourier Descriptors (GFD)
			9.3.2 Generalized Color Fourier Descriptors (GCFD)
		9.4 Color Phase Correlation
			9.4.1 Phase Correlation for Grayscale Images
			9.4.2 Phase Correlation for Color Images
		9.5 Experiments
			9.5.1 Image Database
			9.5.2 Descriptors Extraction
			9.5.3 Classification
			9.5.4 Evaluation of the GCFD
			9.5.5 Evaluation of the Phase Correlation
		9.6 Conclusion
		9.7 Exercises
		References
Part IV: Theorem Proving and Combinatorics
	Chapter 10: On Geometric Theorem Proving  with Null Geometric Algebra
		10.1 Introduction
		10.2 Null Grassmann-Cayley Algebra and Null Bracket Algebra
			10.2.1 Grassmann-Cayley Algebra, Bracket Algebra and Inner-Product Bracket Algebra
			10.2.2 From Conformal Geometric Algebra to Null Grassmann-Cayley Algebra and Null Bracket Algebra
		10.3 Applications: Geometric Factorization, Decomposition, and Theorem Completion
			Input
			Output
			Part 1. Elimination
			Part 2. Homogenization
		10.4 Conclusion
		10.5 Exercises
		References
	Chapter 11: On the Use of Conformal Geometric  Algebra in Geometric Constraint  Solving
		11.1 Declarative Modeling of Geometric Systems
		11.2 Geometric Constraint Solving
		11.3 Topologically and Technologically Related Surfaces (TTRS)
			11.3.1 Lie Algebra of the Group of Rigid Motions
			11.3.2 Geometric Algebra Considerations
		11.4 CGA for Geometric Constraint Solving
			11.4.1 Example of Symbolic Solving
				11.4.1.1 Representation of Geometric Elements
				11.4.1.2 Representation of Geometric Constraints
				11.4.1.3 Symbolic Solving
			11.4.2 Example of Geometric Classification
				11.4.2.1 Representation of Geometric Elements
				11.4.2.2 Geometric Interpretation of TTRS5
				11.4.2.3 Classification of TTRS5
		11.5 Open Problems
			11.5.1 Chirality Specification
			11.5.2 Mobility Specification
		11.6 Conclusion
		11.7 Exercises
		References
	Chapter 12: On the Complexity of Cycle Enumeration for Simple Graphs
		12.1 Introduction
		12.2 Essential Background
		12.3 Technical Considerations
		12.4 Theoretical Complexity
			12.4.1 Average-Case Complexity in "Suitably Sparse" Graphs
		12.5 Implementation Notes
		12.6 Conclusion
		12.7 Exercises
		References
Part V: Applications of Line Geometry
	Chapter 13: Line Geometry in Terms of the Null Geometric Algebra over R3,3,  and Application to the Inverse Singularity Analysis of Generalized Stewart Platforms
		13.1 Introduction
		13.2 Line Geometry with Null Geometric Algebra
		13.3 Inverse Singularity Analysis by Wrench Matrix
		13.4 Singular Configurations of GSPs
			Singularity type 1
			Singularity type 2
			Singularity type 3
			Singularity 4
		13.5 Conclusion
		13.6 Exercises
		References
	Chapter 14: A Framework for n-Dimensional Visibility Computations
		14.1 Problem Statement
			14.1.1 About Visibility
			14.1.2 The Dimension Problem
			14.1.3 Toward a Global Visibility Framework
		14.2 Line Spaces
			14.2.1 n-Dimensional Lines
			14.2.2 From Line to Line Space
			14.2.3 About the Grassmannian
			14.2.4 Line Orientation
			14.2.5 Dual Line Representation
		14.3 Visibility in Ln
			14.3.1 Lines Stabbing a Convex (n-1)-Face
			14.3.2 Convex Cells and Visibility Events in the Line-Space
				14.3.2.1 Interpretation in Ln and Consequences
				14.3.2.2 Global Visibility in Gn as Convex Cells in Ln
				14.3.2.3 Visibility Events in Ln
		14.4 The Minimal Polytope
			14.4.1 Minimal Polytope Interest
			14.4.2 The Minimal Polytope for Two Convex Faces
			14.4.3 Proof of the Minimal Polytope Solution
				14.4.3.1 If the Polytope MAB Exists, then It Is Minimal
				14.4.3.2 Proof of SABMAB
				14.4.3.3 Proof of MABGR(2,n+1) SAB
				14.4.3.4 When the Hyperplane HB or HA Intersects the Inside of A or B
				14.4.3.5 Dealing with Degenerate Cases
		14.5 An Application Example: Soft Shadows Computation
			14.5.1 The n-Dimensional Visibility Framework Implementation
			14.5.2 Soft Shadow Computations
				14.5.2.1 Application Overview
				14.5.2.2 Results
		14.6 Exercises
		References
Part VI: Alternatives to Conformal Geometric Algebra
	Chapter 15: On the Homogeneous Model of Euclidean Geometry
		15.1 Introduction
		15.2 The Grassmann Algebra(s) of Projective Space
			Grassmann Algebra
			Simple and Nonsimple Vectors
			Projectivized Exterior Algebra
			Dual Exterior Algebra
			15.2.1 Remarks on Homogeneous Coordinates
			15.2.2 Equal Rights for W and W*
				Bases and Isomorphisms for W and W*
				The Isomorphism J
				The Canonical Basis
				Projective Join and Meet
				There Are No Lines, Only Spears and Axes!
		15.3 Clifford Algebra for Euclidean Geometry
			15.3.1 The Cayley-Klein Construction
				Polarity on the Metric Quadric
				Free Vectors and the Euclidean Metric
			15.3.2 A Model for Euclidean Geometry
				Counterspace
			15.3.3 J, Metric Polarity, and the Regressive Product
		15.4 The Euclidean Plane via P(R*2,0,1)
			Consequences of Degeneracy
			Notation
			15.4.1 Enumeration of Various Products
			15.4.2 Euclidean Isometries via Sandwich Operations
				Reflections
				Direct Isometries
				Rotations
			15.4.3 Spin Group, Exponentials, and Logarithms
				Lie Groups and Lie Algebras
			15.4.4 Guide to the Literature
		15.5 P(R*3,0,1) and Euclidean Space
			Notation
			15.5.1 Properties of Bivectors
				Null System
				Metric Properties of Bivectors
				Guide to the Literature
			15.5.2 Enumeration of Various Products
			15.5.3 Dual Numbers
				Dual Analysis
				The Axis of a Bivector
			15.5.4 Reflections, Translations, Rotations, and …
			15.5.5 Rotors, Exponentials and Logarithms
		15.6 Case Study: Rigid Body Motion
			15.6.1 Kinematics
				Null Plane Interpretation
			15.6.2 Dynamics
				3D Statics
				15.6.2.1 Newtonian Particles
					Remarks
					Inertia Tensor of a Particle
				15.6.2.2 Rigid Body Motion
					Clifford Algebra for Inertia Tensor
				15.6.2.3 The Euler Equations for Rigid Body Motion
				15.6.2.4 Solving for the Motion
					Comparison
		15.7 Guide to the Literature
		15.8 The Homogeneous Model: A Serious Alternative
		15.9 Non-Euclidean Extension
		15.10 Conclusion
		15.11 Exercises
		References
	Chapter 16: A Homogeneous Model  for Three-Dimensional  Computer Graphics Based  on the Clifford Algebra for R3
		16.1 Introduction
		16.2 The Standard Model of the Clifford Algebra for Three Dimensions
		16.3 Operands and Operators: Mass-Points and Quaternions
			16.3.1 Odd Order: Mass-Points
			16.3.2 Even Order: Quaternions
		16.4 Decomposing Mass-Points into Two Complementary Planes
			16.4.1 Action of q(b,theta) on b||
			16.4.2 Action of q(b,theta) on b
			16.4.3 Sandwiching
		16.5 Rotation, Reflection, and Perspective Projection
			16.5.1 Rotation
			16.5.2 Mirror Image
			16.5.3 Perspective Projection
		16.6 Summary
		16.7 Exercises
		References
	Chapter 17: Rigid-Body Transforms Using Symbolic Infinitesimals
		17.1 Introduction
		17.2 Geometric Algebra G4
		17.3 Geometry and Transforms
		17.4 Rotations and Translations
		17.5 Motions
		17.6 Conclusions
		17.7 Exercises
		References
	Chapter 18: Rigid Body Dynamics in a Constant  Curvature Space and the `1D-up' Approach  to Conformal Geometric Algebra
		18.1 Introduction
		18.2 The `1D up' approach
			18.2.1 Equations and Solutions for Rigid Body Motion in Spherical Space
				18.2.1.1 Point Particle Motion in Curved Space
				18.2.1.2 Rigid Body Motion in Curved Space
				18.2.1.3 The Dumbbell Motion
		18.3 Comparison with Charles Gunn's Work on Euclidean Rigid Body Motion
			18.3.1 Translation into CGA
			18.3.2 Applications to the Euclidean Model and Rigid Bodies
			18.3.3 Comparison with the Curved Space Approach
		18.4 Conclusions
		18.5 Exercises
		References
Part VII: Towards Coordinate-Free Differential Geometry
	Chapter 19: The Shape of Differential Geometry in Geometric Calculus
		19.1 Introduction
		19.2 Geometric Calculus-Basic Concepts
		19.3 Differentiable Manifolds as Vector Manifolds
		19.4 Directed Integrals and the Fundamental Theorem
		19.5 Mappings and Transformations
		19.6 Shape and Curvature
		19.7 Hypersurfaces and Classical Geometry
		19.8 Challenges
		19.9 Exercises
		References
	Chapter 20: On the Modern Notion of a Moving  Frame
		20.1 Introduction
		20.2 Invariants
			20.2.1 Differential Invariants and Their Syzygies
			20.2.2 Integral Invariants
			20.2.3 Joint Invariants
		20.3 Moving Frames
			20.3.1 The Definition of a Moving Frame
			20.3.2 The Calculation of a Moving Frame
				20.3.2.1 A Frame for the Action on Derivatives
				20.3.2.2 A Frame for the Joint Action
		20.4 Invariants via Moving Frames
			20.4.1 Joint Invariants via Moving Frames
			20.4.2 Differential Invariants via Moving Frames
			20.4.3 Moving Frames for Integral Invariants
		20.5 Moving Frames for the SE(3) Action in Conformal Geometric Algebra
			20.5.1 The Serret-Frenet Frame in CGA
			20.5.2 Going Co-ordinate Free
		20.6 Exercises
		References
	Chapter 21: Tutorial Appendix: Structure Preserving Representation of Euclidean Motions  Through Conformal Geometric Algebra
		21.1 Introduction
		21.2 Conformal Geometric Algebra
			21.2.1 Trick 1: Representing Euclidean Points in Minkowski Space
			21.2.2 Trick 2: Orthogonal Transformations as Multiple Reflections in a Sandwiching Representation
			21.2.3 Trick 3: Constructing Elements by Anti-symmetry
			21.2.4 Trick 4: Dual Specification of Elements Permits Intersection
		21.3 Bonus: The Elements of Euclidean Geometry as Blades
		21.4 Bonus: Rigid Body Motions Through Sandwiching
		21.5 Bonus: Structure Preservation and the Transfer Principle
		21.6 Trick 5: Exponential Representation of Versors
		21.7 Trick 6: Geometric Calculus
		21.8 Trick 7: Sparse Implementation at Compiler Level
		21.9 Exercises
		References
Index