Applications of Geometric Algebra in Computer Science and Engineering

by Dorst, Leo; Doran, Chris; Lasenby, Joan

• ISBN13: 9780817642679
• ISBN10: 0817642676
• Format: Hardcover
• Copyright: 2002-03-01
• Publisher: Birkhauser

Summary

Geometric algebra has established itself as a powerful and valuable mathematical tool for solving problems in computer science, engineering, physics, and mathematics. The articles in this volume, written by experts in various fields, reflect an interdisciplinary approach to the subject, and highlight a range of techniques and applications. Relevant ideas are introduced in a self-contained manner and only a knowledge of linear algebra and calculus is assumed. Features and Topics: * The mathematical foundations of geometric algebra are explored * Applications in computational geometry include models of reflection and ray-tracing and a new and concise characterization of the crystallographic groups * Applications in engineering include robotics, image geometry, control-pose estimation, inverse kinematics and dynamics, control and visual navigation * Applications in physics include rigid-body dynamics, elasticity, and electromagnetism * Chapters dedicated to quantum information theory dealing with multi-particle entanglement, MRI, and relativistic generalizations Practitioners, professionals, and researchers working in computer science, engineering, physics, and mathematics will find a wide range of useful applications in this state-of-the-art survey and reference book. Additionally, advanced graduate students interested in geometric algebra will find the most current applications and methods discussed.

Table of Contents

Preface	v
Contributors	vii
I Algebra and Geometry	1	(194)
Point Groups and Space Groups in Geometric Algebra	3	(32)
David Hestenes
Introduction	3	(2)
Point Groups in Two Dimensions	5	(5)
Point Groups in Three Dimensions	10	(7)
The 32 Crystal Classes and 7 Crystal Systems	17	(5)
Homogeneous Euclidean Geometry	22	(3)
Symmetries from Reflections	25	(2)
The Space Groups	27	(8)
Planar Space Groups	29	(3)
3d Space Groups	32	(3)
The Inner Products of Geometric Algebra	35	(12)
Leo Dorst
The Product Structure of Geometric Algebra	35	(1)
The Basic Products of Geometric Algebra	36	(4)
Geometric Product	36	(1)
Outer Product; Blades	37	(1)
Scalar Product: Metric Properties	37	(1)
Contractions	38	(1)
Relationship of the Contraction to the Inner Product	39	(1)
Understanding the Contraction	40	(2)
Defining Axioms	40	(1)
Famous Formulas	40	(1)
Geometric Interpretation	41	(1)
Projection	42	(1)
Meet and Join	43	(1)
Linear Transformations as `Innermorphisms'	44	(1)
Outermorphisms and Adjoint Transformations	44	(1)
Covariance of Inner Product Formulas	44	(1)
Replacing the Inner Product	45	(2)
Unification of Grassmann's Progressive and Regressive Products using the Principle of Duality	47	(12)
Stephen Blake
Introduction	47	(1)
Points	48	(1)
Grassmann's Algebra of Points	49	(2)
Hyperplanes and Coordinates for Points	51	(1)
Points and Coordinates for Hyperplanes	51	(1)
Representing Regions by Points or Hyperplanes	52	(1)
Rule of the Middle Factor	53	(1)
Fundamental Formulae of Whitehead's Algebra	54	(1)
Example: The Harmonic Section Theorem	55	(4)
From Unoriented Subspaces to Blade Operators	59	(10)
Timaeus A. Bouma
Introduction	59	(2)
A Review of Geometric Algebra	59	(2)
Definitions	61	(2)
Projection Operators	61	(1)
Delta Product	61	(1)
LIFT	62	(1)
Meet and Join	62	(1)
Projection Operators	63	(1)
Projection and the Inner Product	63	(1)
Fundamental Theorem of Projection Operators	63	(1)
Special Cases	63	(1)
The Meet and Join for Projection Operators	64	(1)
The Meet and Join When AB Is a Blade	64	(1)
The Meet and Join When AB Is Not a Blade	64	(1)
From Unoriented Subspaces to Blade Operators	65	(1)
The Blade Correspondence	65	(1)
LIFT to a Euclidean Metric	66	(1)
Conclusion	66	(3)
Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra	69	(10)
Hongbo Li
Clifford Bracket Algebra	71	(1)
Some Bracket Expressions	72	(2)
Theorem Proving with Clifford Bracket Algebra	74	(5)
Rotations in n Dimensions as Spherical Vectors	79	(12)
W. E. Baylis
S. Hadi
Introduction	79	(2)
Sn-1 Model	81	(6)
Spinor Rotations and Spherical Vectors	81	(3)
Composing Rotations by Adding Spherical Vectors	84	(3)
Examples	87	(2)
Rotations in E3	87	(1)
Rotations in E4	88	(1)
Conclusions	89	(2)
Geometric and Algebraic Canonical Forms	91	(8)
Neil Gordon
Introduction	91	(1)
Clifford Algebras and Finite Geometry	91	(2)
Geometric Structures and Algebraic Forms	93	(2)
Geometric Canonical Forms, Algebraic Canonical Forms and Computer Algebra	95	(1)
Conclusion	96	(3)
Functions of Clifford Numbers or Square Matrices	99	(10)
John Snygg
Introduction	99	(1)
Using the Minimal Polynomial	99	(5)
Using the Characteristic Polynomial	104	(2)
Concluding Remarks	106	(3)
Compound Matrices and Pfaffians: A Representation of Geometric Algebra	109	(10)
Uwe Prells
Michael. I. Friswell
Seamus D. Garvey
Grassmann Algebra and Compound Matrices	109	(3)
Pfaffians and their Generalisation	112	(2)
Representation of the Clifford Algebra Cln(F)	114	(2)
Clifford Powers	116	(3)
Analysis Using Abstract Vector Variables	119	(10)
Frank Sommen
Introduction	119	(1)
The Algebra of Abstract Vector Variables	119	(4)
Analysis in Abstract Vector Variables	123	(4)
Conclusion	127	(2)
A Multivector Data Structure for Differential Forms and Equations	129	(4)
Jeffrey A. Chard
Vadim Shapiro
Jet Bundles and the Formal Theory of Partial Differential Equations	133	(12)
Richard Baker
Chris Doran
Fibre Bundles and Sections	133	(1)
Jet Bundles	134	(1)
Differential Functions and Formal Derivatives	135	(1)
Prolongation of Sections	135	(1)
Differential Equations and Solutions	136	(1)
Prolongation and Projection of Differential Equations	137	(1)
Power Series Solutions	137	(1)
Integrability Conditions	138	(1)
Involutive Differential Equations	139	(2)
Cartan--Kuranishi Completion	141	(1)
Conclusion	142	(3)
Imaginary Eigenvalues and Complex Eigenvectors Explained by Real Geometry	145	(12)
Eckhard M. S. Hitzer
Introduction	145	(1)
Two Real Dimenstions	146	(3)
Complex Treatment	146	(1)
Real Explanation	147	(2)
Three Real Dimensions	149	(4)
Complex Treatment of Three Dimensions	149	(1)
Real Explanation for Three Dimensions	150	(3)
Four Euclidean Dimensions --- Complex Treatment	153	(4)
Symbolic Processing of Clifford Numbers in C++	157	(12)
John P. Fletcher
Introduction	157	(1)
SymbolicC++	158	(3)
Operator and Type Definitions	158	(1)
Type Conversion	159	(1)
Output	160	(1)
Noncommutative Arithmetic	160	(1)
Extending SymbolicC++	160	(1)
Clifford Algebra Examples	161	(3)
Quaternion Starting Point	161	(1)
Clifford (2)	161	(2)
Clifford (3)	163	(1)
Clifford (2,2)	163	(1)
Interactive Interface to Tel using SWIG	164	(1)
Storing Algebra using XML and E4Graph	165	(1)
Conclusions	166	(3)
Clifford Numbers and their Inverses Calculated using the Matrix Representation	169	(10)
John P. Fletcher
Introduction	169	(1)
Clifford Basis Matrix Theory	170	(2)
Calculation of the Inverse of a Clifford Number	172	(6)
Example 1: Clifford (2)	173	(1)
Example 2: Clifford (3)	173	(2)
Example 3: Clifford (2,2)	175	(3)
Conclusion	178	(1)
A Toy Vector Field Based on Geometric Algebra	179	(8)
Alyn Rockwood
Shoeb Binderwala
Introduction	179	(2)
The Conjugate Field Method	181	(2)
Comparison	183	(1)
Implementation	184	(3)
Quadratic Transformations in the Projective Plane	187	(6)
Georgi Georgiev
Introduction	187	(1)
Quadratic Involution in the Projective Plane	187	(2)
Quadratic Involution which Leaves an Invariant Ellipse	189	(4)
Annihilators of Principal Ideals in the Grassmann Algebra	193	(2)
Cemal Koc
Songul Esin
II Applications to Physics	195	(122)
Homogeneous Rigid Body Mechanics with Elastic Coupling	197	(16)
David Hestenes
Ernest D. Fasse
Introduction	197	(2)
Homogeneous Euclidean Geometry	199	(3)
Rigid Displacements	202	(4)
Kinematics	206	(2)
Dynamics	208	(1)
Elastic Coupling	209	(1)
Conclusions	210	(3)
Analysis of One and Two Particle Quantum Systems using Geometric Algebra	213	(14)
Rachel Parker
Chris Doran
Introduction	213	(1)
Single-Particle Pure States	214	(1)
2-Particle Systems	214	(3)
Schmidt Decomposition	215	(1)
The Density Matrix	216	(1)
Geometric Algebra	217	(10)
Single-Particle Systems	217	(2)
2-Particle Systems	219	(3)
2-Particle Observables	222	(1)
The Density Matrix	223	(1)
Example - The Singlet State	224	(3)
Interaction and Entanglement in the Multiparticle Spacetime Algebra	227	(22)
Timothy F. Havel
Chris J. L. Doran
The Physics of Quantum Information	227	(2)
The Multiparticle Spacetime Algebra	229	(1)
Two Interacting Qubits	230	(7)
The Propagator	233
Observables	34	(203)
Lagrangian Analysis	237	(3)
Single-Particle Systems	237	(1)
Two-Particle Interactions	238	(1)
Symmetries and Noether's theorem	239	(1)
The Density Operator	240	(9)
An Example of Information Dynamics	242	(2)
Towards Quantum Complexity and Decoherence	244	(5)
Laws of Reflection from Two or More Plane Mirrors in Succession	249	(12)
Mike Derome
Introduction	249	(1)
Single Reflection in 3D Euclidean Space	250	(1)
Two Successive Reflections	251	(2)
Definitions	252	(1)
Laws of Double Reflection	252	(1)
The Spread Angle for Two Reflections	253	(1)
Persistence for Two Reflections	253	(1)
Reflection in the Plane of Mirror Normals - 2D Reflection	253	(1)
Retro-reflection Solutions for Two Reflections	253	(1)
Three Successive Reflections	253	(4)
Two Important Special Configurations Exist:	254	(1)
Laws of Triple Reflections	255	(1)
The Spread Angle for Three Reflections	255	(1)
Persistence for Three Reflections	256	(1)
Reflection in the Plane of all Mirror Normals - 2D Reflection	256	(1)
Retro-reflective Solutions for Three Reflections	256	(1)
Constraints on Retro-reflection from Three One-sided Mirrors	256	(1)
An Arbitrary Number of Successive Reflections	257	(1)
Conclusions	258	(3)
Exact Kinetic Energy Operators for Polyatomic Molecules	261	(10)
Janne Pesonen
Introduction	261	(1)
Kinetic Energy Operator	262	(6)
Vibrational Degrees of Freedom	264	(1)
Rotational Degrees of Freedom	265	(3)
Comparison to other Approaches	268	(3)
Geometry of Quantum Computing by Hamiltonian Dynamics of Spin Ensembles	271	(14)
T. Schulte-Herbruggen
K. Huper
U. Helmke
S. J. Glaser
From Hamiltonian Quantum Dynamics to C-Numerical Ranges	271	(3)
Quantum Mechanics of Elementary and Joint Systems	271	(1)
Reduced Descriptions of Ensembles	272	(1)
Relation to the C-Numerical Range	273	(1)
Unitary Controllability of Spin Ensembles	274	(2)
From Geometry to Entropy and vice versa	276	(3)
Generalised Angles between States or Matrices	276	(1)
Euclidean Distances between States or Matrices	277	(1)
Entropies vs Norms and Euclidean Distances	277	(2)
Gradient Flow Minimising Euclidean Distance	279	(1)
A Caveat on Entropy Measures in NMR	280	(5)
Is the Brain a 'Clifford Algebra Quantum Computer'?	285	(12)
V. Labunets
E. Rundblad
J. Astola
Introduction	285	(2)
Clifford Algebra as a Model of Geometrical Space	287	(1)
Clifford Algebra as a Model of Perceptual Space	288	(3)
Hypercomplex-valued Invariants	291	(1)
Fast Calculation Algorithms	292	(5)
A Hestenes Spacetime Algebra Approach to Light Polarization	297	(10)
Qurino M. Sugon
Daniel McNamara
Introduction	297	(1)
Electromagnetic Wave	298	(2)
The Energy-Momentum Cliffor	300	(1)
The Poincare Sphere, Stokes Parameters and Coherency Matrices	301	(2)
Polarization States	303	(2)
Summary	305	(2)
Quaternions, Clifford Algebra and Symmetry Groups	307	(10)
Patrick R. Girard
Introduction	307	(1)
Clifford Algebras	307	(2)
Definitions and Theorem	307	(1)
Clifford Algebra (C4 over R)	308	(1)
The Lorentz Group	309	(2)
Pseudo-euclidean Space	309	(1)
Riemannian Space	310	(1)
The Conformal Group	311	(1)
Dirac Algebra (C4 over C)	312	(1)
Dirac's Equation	312	(1)
Unitary and Symplectic Groups	312	(1)
Conclusion	313	(4)
III Computer Vision and Robotics	317	(106)
A Generic Framework for Image Geometry	319	(14)
Jan J. Koenderink
Introduction	319	(1)
The Natural Intensity Scale	320	(1)
The Similarities of Image Space	321	(1)
Geometries of the Picture and Normal Planes Compared	322	(2)
The Differential Geometry of the Normal Planes	324	(1)
Geometry of Image Space	324	(1)
Differential Geometry of Images	325	(4)
Curves	326	(1)
Surfaces	327	(2)
Relation with Scale Space Structure	329	(1)
Conclusions	329	(4)
Color Edge Detection Using Rotors	333	(8)
Eduardo Bayro-Corrochano
Sandino Flores
Introduction	333	(2)
Rotors	334	(1)
Rotor Edge Detector	335	(1)
Modified Rotor Edge Detector	336	(3)
Conclusion	339	(2)
Numerical Evaluation of Versors with Clifford Algebra	341	(10)
Christian B. U. Perwass
Gerald Sommer
Introduction	341	(1)
Theory	342	(4)
Implementation	346	(1)
Experiments	347	(2)
Conclusion	349	(2)
The Role of Clifford Algebra in Structure-Preserving Transformations for Second-Order Systems	351	(10)
Seamus D. Garvey
Michael I. Friswell
Uwe Prells
The Characteristic Behaviour of Second-Order Systems	351	(1)
Initial Indications of a Role for Cl2	352	(4)
Structure Preserving Transformations for Second-Order Systems	356	(5)
Applications of Algebra of Incidence in Visually Guided Robotics	361	(12)
Eduardo Bayro-Corrochano
Pertti Lounesto
Leo Reyes Lozano
Algebra of Incidence	361	(5)
Incidence Relations in the Affine n-plane	362	(2)
Incidence Relations in the Affine 3-plane	364	(1)
Geometric Constraints as Indicators	365	(1)
Rigid Motion in the Affine Plane	366	(1)
Application to Robotics	367	(6)
Inverse Kinematic Computing	367	(3)
Robot Manipulation Guidance	370	(1)
Checking for a Critical Configuration	371	(2)
Monocular Pose Estimation of Kinematic Chains	373	(12)
Bodo Rosenhahn
Oliver Granert
Gerald Sommer
Introduction	373	(1)
Pose Estimation in Conformal Geometric Algebra	374	(3)
The Scenario of Pose Estimation	374	(1)
Introduction to Conformal Geometric Algebra	375	(1)
Kinematic Constraints in Conformal Geometric Algebra	376	(1)
Pose Estimation of Kinematic Chains	377	(2)
Kinematic Chains in Conformal Geometric Algebra	377	(1)
Constraint Equations for Kinematic Chains	378	(1)
Experiments	379	(2)
Discussion	381	(4)
Stabilization of 3D Pose Estimation	385	(10)
W. Neddermeyer
M. Schnell
W. Winkler
A. Lilienthal
Introduction	385	(1)
Camera Model	385	(2)
Determination of Position	387	(2)
Determination of Position by the Bundle Adjustment Method	387	(1)
Reference Measurement	388	(1)
Stabilization	389	(3)
Stabilization of the Operating Point by Variation of Camera Position	389	(2)
Stabilization of the Operating Point by Variation of the Object Parameters	391	(1)
Example	392	(1)
Concluding Remarks	392	(3)
Inferring Dynamical Information from 3D Position Data using Geometric Algebra	395	(12)
Hiniduma Udugama Gamage Sahan Sajeewa
Joan Lasenby
Introduction	395	(1)
Some Basic Formulations	396	(2)
Angular Velocity	396	(1)
Linear Velocity, Acceleration and Inertial Force	397	(1)
Angular Momentum, Inertia Tensor and Inertial Torque	397	(1)
Calculations in Terms of Rotational Bivectors	398	(1)
Algorithm for Inverse Dynamics	399	(2)
Dynamical Equilibrium in the Model	401	(1)
Inverse Dynamics from Motion Capture Data	402	(1)
Real World Applications and Results	403	(2)
Conclusions and Future Work	405	(2)
Clifford Algebra Space Singularities of Inline Planar Platforms	407	(16)
Michael A. Baswell
Rafat Abtamowicz
Joe N. Anderson
Introduction	407	(2)
VGT Singularities	408	(1)
Paper Organization	409	(1)
Singularity Types and Clifford Algebras	409	(4)
Type I, Type II, and Type III Singularities	409	(1)
The Clifford Algebra Cl+(P3)	410	(1)
Cl+(P3) Components from the Screw Parameterization	411	(2)
Singularity Surfaces of Single Stage Planar Platforms	413	(5)
The Homogeneous Transformations	413	(1)
The Constraint Manifold	413	(2)
The Quaternionic Jacobian	415	(1)
Singularity Sets of Planar Platforms	416	(2)
Singularity Surfaces of Two Stacked Planar Platforms	418	(2)
The Jacobian of Two Stacked General Inline Planar Platforms	419	(1)
Conclusions and Recommendations	420	(3)
IV Signal Processing and Other Applications	423	(52)
Fast Quantum Fourier--Heisenberg--Weyl Transforms	425	(12)
V. Labunets
E. Rundblad
J. Astola
Introduction	425	(2)
Heisenberg--Weyl Groups	427	(2)
Quantum Fourier--Heisenberg--Weyl Transform	429	(8)
The Structure Multivector	437	(12)
Michael Felsberg
Gerald Sommer
Introduction and Motivation	437	(3)
Mathematical Fundamentals	440	(2)
The Approach for i1D Symmetries	442	(1)
The Approach for i2D Symmetries	443	(1)
The Structure Multivector	444	(2)
Conclusion	446	(3)
The Application of Clifford Algebra to Calculations of Multicomponent Chemical Composition	449	(10)
John P. Fletcher
Introduction	449	(1)
Projection	450	(1)
The V Product	451	(1)
Multicomponent Chemical Composition	451	(5)
Projection Model	451	(2)
Conversion between Mass and Molar Basis	453	(1)
Vapour-Liquid Equilibrium	454	(1)
Flash Calculation	455	(1)
Chemical Reaction	455	(1)
Volume and Thermodynamic Functions	456	(1)
Vee Product Model	456	(2)
Generalisation	458	(1)
Conclusions	458	(1)
An Algorithm to Solve the Inverse IFS-Problem	459	(10)
Erwin Hocevar
Introduction	459	(1)
Problem Specification --- Objects to be Encoded	459	(1)
Algorithm to Calculate the IFS-Codes of an Image	460	(3)
Calculating Boundaries of an Object	460	(1)
Calculating Orbits Defining a Not Minimal IFS	461	(1)
Calculating a Minimal IFS	462	(1)
Calculating IFS-Codes for the Inner Image Parts	462	(1)
Conclusion	463	(6)
Summary	463	(1)
Future Work	464	(5)
Fast Quantum n-D Fourier and Radon Transforms	469	(6)
V. Labunets
E. Rundblad
J. Astola
Introduction	469	(1)
Discrete Radon and K-Transforms	469	(2)
Fast Classical Random and K-Transforms	471	(4)
Fast 2D Randon and K-Transforms	471	(2)
Fast nD K-Transform on Zpn-1 x MZp	473	(1)
Fast nD K-Transform on the Zpmn-1 x MZpm	474	(1)
Fast Quantum Fourier, Randon and K-Transforms	475