EDITORS EDUARDO BAYRO CORROCHANO and GARRET SOBCZYK Birkhauser, Boston, 2001 ISBN 0-8176-4199-8 Preface Part I Advances in Geometric Algebra Chapter 1 Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry, David Hestenes 1.1 Introduction 1.2 Minkowski Algebra 1.3 Conformal Split 1.4 Models of Euclidean Space 1.5 Lines and Planes 1.6 Spheres and Hyperplanes 1.7 Conformal and Euclidean Groups 1.8 Screw Mechanics 1.9 Conclusions Chapter 2 Universal Geometric Algebra Garret Sobczyk 2.1 Introduction 2.2 The Universal Geometric Algebra 2.3 Matrices of Geometric Numbers 2.4 Linear Transformations 2.5 Pseudo-Euclidean Geometries 2.6 Affine and Projective Geometries 2.7 Conformal Transformations Chapter 3 Realizations of the Conformal Group Jose Maria Pozo and Garret Sobczyk 3.1 Introduction 3.2 Projective Geometry 3.3 The Conformal Representant and Stereographic Projection 3.4 Conformal Transformations and Isometries 3.5 Isometries in No 3.6 Compactification 3.7 Mobius Transformations Chapter 4 Hyperbolic Geometry Hongbo Li 4.1 Introduction 4.2 Hyperbolic Plane Geometry with Clifford Algebra 4.3 Hyperbolic Conformal Geometry with Clifford Algebra 4.4 A Universal Model for the Conformal Geometries of the Euclidean, Spherica and Double-Hyperbolic Spaces 4.5 Conclusion Part II Theorem Proving Chapter 5 Geometric Reasoning With Geometric Algebra Dongming Wang 5.1 Introduction 5.2 Clifford Algebra for Euclidean Geometry 5.4 Proving Identities in Clifford Algebra Chapter 6 Automated Theorem Proving Hongbo Li 6.1 Introduction 6.2 A general Framework for Clifford algebra and Wu's Method 6.3 Automated Theorem Proving in Euclidean Geometry and Other Classical Geometries 6.4 Automated Theorem Proving in Differential Geometry}{116} 6.5 Conclusion Part III Computer Vision Chapter 7 The Geometry Algebra of Computer Vision Eduardo Bayro Corrochano and Joan Lasenby 7.1 Introduction 7.2 The Geometric Algebras of 3-D and 4-D Spaces 7.3 The Algebra of Incidence 7.4 Algebra in Projective Space 7.5 Visual Geometry of $n$ Uncalibrated Camera 7.6 Conclusions Chapter 8 Using Geometric Algebra for Optical Motion Capture Joan Lasenby and Adam Stevenson 8.1 Introduction 8.2 External and Internal Calibration 8.3 Estimating the External Parameters 8.4 Examples and Results 8.5 Extending to Include Internal Calibration 8.6 Conclusions Chapter 9 Bayesian Inference and Geometric Algebra: An Application to Camera Localization Chris Doran 9.1 Introduction 9.2 Geometric Algebra in Three Dimensions 9.3 Rotors and Rotations 9.4 Rotor Calculus 9.5 Computer Vision 9.6 Unknown range data 9.7 Extension to three cameras 9.8 Conclusions Chapter 10 Projective Reconstruction of Shape and Motion Using Invariant Theory Eduardo Bayro Corrochano and Vladimir Banarer 10.1 Introduction 10.2 3-D Projective Invariants from Multiple Views 10.3 Projective Depth 10.4 Shape and Motion 10.5 Conclusions Part IV Robotics Chapter 11 Robot Kinematics and Flags 11.1 Introduction 11.2 The Clifford Algebra 11.3 Flags 11.4 Robots 11.5 Concluding Remarks Chapter 12 The Clifford Algebra and the Optimization of Robot Design Shawn G. Ahlers and John Michael McCarthy 12.1 Introduction 12.2 Literature Review 12.3 Overview of the Design Algorithm 12.4 Double Quaternions 12.5 The Task Trajectory 12.6 The Design of the TS Robot 12.7 The Optimum TS Robot 12.8 Conclusion Chapter 13 Eduardo Bayro Corrochano and Garret Sobczyk 13.1 Introduction 13.2 The General Linear Group 13.3 Algebra of Incidence 13.4 Rigid Motion in the Affine Plane 13.5 Application to Robotics 13.6 Application II: The design of an image filter Recognition of hand gestures The meet filter 13.7 Conclusion Part V Quantum and Neural Computing, and Wavelets Chapter 14 Geometric Algebra in Quantum Information Processing by Nuclear Magnetic Resonance Timothy F. Havel, David G. Cory, Shyamal S. Somaroo, and Ching-Hua Tseng 14.1 Introduction 14.2 Multiparticle Geometric Algebra 14.3 Algorithms for Quantum Computers 14.4 NMR and the Product Operator Formalism 14.5 Quantum Computing by Liquid-State NMR 14.6 States and Gates by NMR 14.7 Quantum Simulation by NMR 14.8 Remarks on Foundational Issues Chapter 15 Geometric Feedforward Neural Networks and Support Multivector Machines Eduardo Bayro Corrochano and Refugio Vallejo 15.1 Introduction 15.2 Real Valued Neural Networks 15.3 Complex MLP and Quaternionic MLP 15.4 Geometric Algebra Neural Networks 15.5 Learning Rule 15.6 Experiments Using Geometric Feedforward Neural Networks 15.7 Support Vector Machines in Geometric Algebra 15.8 Experimental Analysis of Support Multivector Machines 15.9 Conclusions Chapter 16 Image Analysis Using Quaternion Wavelets Leonardo Traversoni 16.1 Introduction 16.2 The Static Approach 16.3 Clifford Multiresolution Analyses 16.4 Haar Quaternionic Wavelets 16.5 A Dynamic Interpretation 16.6 Global Interpolation 16.7 Dealing with Trajectories 16.8 Conclusions Part VI Applications to Engineering and Physics Chapte 17 Objects in Contact: Boundary Collisions as Geometric Wave Propagation Leo Dorst 17.1 Introduction 17.2 Boundary Geometry 17.3 The Boundary as a Geometric Object 17.4 Wave Propagation of Boundaries 17.5 Conclusions Chapter 18 Modern Geometric Calculations in Crystallography G. Aragon, J.L. Aragon, F. Davila, A. Gomez and M.A. Rodriguez 18.1 Introduction 18.2 Quasicrystals 18.3 The Morphology of Icosahedral Quasicrystals 18.4 Coincidence Site Lattice Theory 18.5 Conclusions Chapter 19 Quaternion Optimization Problems in Engineering Ljudmila Meister 19.1 Introduction 19.2 Properties of Quaternions 19.3 Extremal Problems for Quaternions 19.4 Determination of Rotations 19.5 The Main Problem of Orientation 19.6 Optimal Filtering and Prediction 19.7 Summary Chapter 20 Clifford Algebras in Electrical Engineering William Baylis 20.1 Introduction 20.2 Structure of Cl_3 20.3 Paravector Model of Spacetime 20.4 Using Relativity at Low Speeds 20.5 Relativity at High Speeds 20.6 Conclusions Chapter 21 Applications of Geometric Algebra in Physics and Links With Engineering Anthony Lasenby and Joan Lasenby 21.1 Introduction 21.2 The Spacetime Algebra 21.3 Quantum Mechanics 21.4 Gravity as a Gauge Theory 21.5 A New Representation of 6-d Conformal Space 21.6 Summary and Conclusions Part VII Computational Methods in Clifford Algebras Chapter 22 Clifford Algebras as Projections of Group Algebras Vladimir M. Chernov 22.1 Introduction 22.2 Group Algebras and Their Projection 22.3 Applications 22.4 Conclusion Chapter 23 Counterexamples for Validation and Discovering of New Theorems Pertti Lounesto 23.1 Introduction 23.2 The Role of Counterexamples in Mathematics 23.3 Clifford Algebras: An Outline 23.4 Preliminary Counterexamples in Clifford Algebras 23.5 Counterexamples About Spin Groups 23.6 Counterexamples on the Internet Chapter 24 The Making of GABLE: A Geometric Algebra Learning Environment in Matlab Stephen Mann, Leo Dorst, and Tim Bouma 24.1 Introduction 24.2 Representation of Geometric Algebra 24.3 Inverses 24.4 Meet and Join 24.5 Graphics 24.6 Example: Pappus's Theorem 24.7 Conclusions Chapter 25 Helmstetter Formula and Rigid Motions with CLIFFORD Rafal Ablamowicz 25.1 Introduction 25.2 Verification of the Helmstetter Formula 25.3 Rigid Motions with Clifford Algebras 25.4 Summary References Index