Applications of Geometric Algebra in Computer Science and Engineering

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

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ISBN: 9780817642679 | 0817642676
Cover: Hardcover
Copyright: 3/1/2002

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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.

Preface

Contributors

vii

I Algebra and Geometry

(194)

Point Groups and Space Groups in Geometric Algebra

(32)

David Hestenes

Introduction

(2)

Point Groups in Two Dimensions

(5)

Point Groups in Three Dimensions

(7)

The 32 Crystal Classes and 7 Crystal Systems

(5)

Homogeneous Euclidean Geometry

(3)

Symmetries from Reflections

(2)

The Space Groups

(8)

Planar Space Groups

(3)

3d Space Groups

(3)

The Inner Products of Geometric Algebra

(12)

Leo Dorst

The Product Structure of Geometric Algebra

(1)

The Basic Products of Geometric Algebra

(4)

Geometric Product

(1)

Outer Product; Blades

(1)

Scalar Product: Metric Properties

(1)

Contractions

(1)

Relationship of the Contraction to the Inner Product

(1)

Understanding the Contraction

(2)

Defining Axioms

(1)

Famous Formulas

(1)

Geometric Interpretation

(1)

Projection

(1)

Meet and Join

(1)

Linear Transformations as `Innermorphisms'

(1)

Outermorphisms and Adjoint Transformations

(1)

Covariance of Inner Product Formulas

(1)

Replacing the Inner Product

(2)

Unification of Grassmann's Progressive and Regressive Products using the Principle of Duality

(12)

Stephen Blake

Introduction

(1)

Points

(1)

Grassmann's Algebra of Points

(2)

Hyperplanes and Coordinates for Points

(1)

Points and Coordinates for Hyperplanes

(1)

Representing Regions by Points or Hyperplanes

(1)

Rule of the Middle Factor

(1)

Fundamental Formulae of Whitehead's Algebra

(1)

Example: The Harmonic Section Theorem

(4)

From Unoriented Subspaces to Blade Operators

(10)

Timaeus A. Bouma

Introduction

(2)

A Review of Geometric Algebra

(2)

Definitions

(2)

Projection Operators

(1)

Delta Product

(1)

LIFT

(1)

Meet and Join

(1)

Projection Operators

(1)

Projection and the Inner Product

(1)

Fundamental Theorem of Projection Operators

(1)

Special Cases

(1)

The Meet and Join for Projection Operators

(1)

The Meet and Join When AB Is a Blade

(1)

The Meet and Join When AB Is Not a Blade

(1)

From Unoriented Subspaces to Blade Operators

(1)

The Blade Correspondence

(1)

LIFT to a Euclidean Metric

(1)

Conclusion

(3)

Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra

(10)

Hongbo Li

Clifford Bracket Algebra

(1)

Some Bracket Expressions

(2)

Theorem Proving with Clifford Bracket Algebra

(5)

Rotations in n Dimensions as Spherical Vectors

(12)

W. E. Baylis

S. Hadi

Introduction

(2)

Sn-1 Model

(6)

Spinor Rotations and Spherical Vectors

(3)

Composing Rotations by Adding Spherical Vectors

(3)

Examples

(2)

Rotations in E3

(1)

Rotations in E4

(1)

Conclusions

(2)

Geometric and Algebraic Canonical Forms

(8)

Neil Gordon

Introduction

(1)

Clifford Algebras and Finite Geometry

(2)

Geometric Structures and Algebraic Forms

(2)

Geometric Canonical Forms, Algebraic Canonical Forms and Computer Algebra

(1)

Conclusion

(3)

Functions of Clifford Numbers or Square Matrices

(10)

John Snygg

Introduction

(1)

Using the Minimal Polynomial

(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

(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

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Applications of Geometric Algebra in Computer Science and Engineering

Summary

Table of Contents

Supplemental Materials