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Geometric Algebra ReviewPart I Advances in Geometric AlgebraChapter 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
Geometric Algebra Overview
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