Ebook: Applications of Geometric Algebra in Computer Science and Engineering

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.

---

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.

---

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.
Content:
Front Matter....Pages i-xxv
Front Matter....Pages 1-1
Point Groups and Space Groups in Geometric Algebra....Pages 3-34
The Inner Products of Geometric Algebra....Pages 35-46
Unification of Grassmann’s Progressive and Regressive Products using the Principle of Duality....Pages 47-57
From Unoriented Subspaces to Blade Operators....Pages 59-67
Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra....Pages 69-78
Rotations in n Dimensions as Spherical Vectors....Pages 79-90
Geometric and Algebraic Canonical Forms....Pages 91-98
Functions of Clifford Numbers or Square Matrices....Pages 99-107
Compound Matrices and Pfaffians: A Representation of Geometric Algebra....Pages 109-118
Analysis Using Abstract Vector Variables....Pages 119-128
A Multivector Data Structure for Differential Forms and Equations....Pages 129-131
Jet Bundles and the Formal Theory of Partial Differential Equations....Pages 133-143
Imaginary Eigenvalues and Complex Eigenvectors Explained by Real Geometry....Pages 145-155
Symbolic Processing of Clifford Numbers in C++....Pages 157-167
Clifford Numbers and their Inverses Calculated using the Matrix Representation....Pages 169-178
A Toy Vector Field Based on Geometric Algebra....Pages 179-185
Quadratic Transformations in the Projective Plane....Pages 187-191
Annihilators of Principal Ideals in the Grassmann Algebra....Pages 193-194
Front Matter....Pages 195-195
Homogeneous Rigid Body Mechanics with Elastic Coupling....Pages 197-212
Analysis of One and Two Particle Quantum Systems using Geometric Algebra....Pages 213-226
Front Matter....Pages 195-195
Interaction and Entanglement in the Multiparticle Spacetime Algebra....Pages 227-247
Laws of Reflection from Two or More Plane Mirrors in Succession....Pages 249-259
Exact Kinetic Energy Operators for Polyatomic Molecules....Pages 261-270
Geometry of Quantum Computing by Hamiltonian Dynamics of Spin Ensembles....Pages 271-283
Is the Brain a ‘Clifford Algebra Quantum Computer’?....Pages 285-295
A Hestenes Spacetime Algebra Approach to Light Polarization....Pages 297-306
Quaternions, Clifford Algebra and Symmetry Groups....Pages 307-315
Front Matter....Pages 317-317
A Generic Framework for Image Geometry....Pages 319-332
Color Edge Detection Using Rotors....Pages 333-339
Numerical Evaluation of Versors with Clifford Algebra....Pages 341-350
The Role of Clifford Algebra in Structure-Preserving Transformations for Second-Order Systems....Pages 351-359
Applications of Algebra of Incidence in Visually Guided Robotics....Pages 361-372
Monocular Pose Estimation of Kinematic Chains....Pages 373-383
Stabilization of 3D Pose Estimation....Pages 385-394
Inferring Dynamical Information from 3D Position Data using Geometric Algebra....Pages 395-406
Clifford Algebra Space Singularities of Inline Planar Platforms....Pages 407-421
Front Matter....Pages 423-423
Fast Quantum Fourier-Heisenberg-Weyl Transforms....Pages 425-435
The Structure Multivector....Pages 437-448
The Application of Clifford Algebra to Calculations of Multicomponent Chemical Composition....Pages 449-458
An Algorithm to Solve the Inverse IFS-Problem....Pages 459-467
Fast Quantum n-D Fourier and Radon Transforms....Pages 469-478

---

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.
Content:
Front Matter....Pages i-xxv
Front Matter....Pages 1-1
Point Groups and Space Groups in Geometric Algebra....Pages 3-34
The Inner Products of Geometric Algebra....Pages 35-46
Unification of Grassmann’s Progressive and Regressive Products using the Principle of Duality....Pages 47-57
From Unoriented Subspaces to Blade Operators....Pages 59-67
Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra....Pages 69-78
Rotations in n Dimensions as Spherical Vectors....Pages 79-90
Geometric and Algebraic Canonical Forms....Pages 91-98
Functions of Clifford Numbers or Square Matrices....Pages 99-107
Compound Matrices and Pfaffians: A Representation of Geometric Algebra....Pages 109-118
Analysis Using Abstract Vector Variables....Pages 119-128
A Multivector Data Structure for Differential Forms and Equations....Pages 129-131
Jet Bundles and the Formal Theory of Partial Differential Equations....Pages 133-143
Imaginary Eigenvalues and Complex Eigenvectors Explained by Real Geometry....Pages 145-155
Symbolic Processing of Clifford Numbers in C++....Pages 157-167
Clifford Numbers and their Inverses Calculated using the Matrix Representation....Pages 169-178
A Toy Vector Field Based on Geometric Algebra....Pages 179-185
Quadratic Transformations in the Projective Plane....Pages 187-191
Annihilators of Principal Ideals in the Grassmann Algebra....Pages 193-194
Front Matter....Pages 195-195
Homogeneous Rigid Body Mechanics with Elastic Coupling....Pages 197-212
Analysis of One and Two Particle Quantum Systems using Geometric Algebra....Pages 213-226
Front Matter....Pages 195-195
Interaction and Entanglement in the Multiparticle Spacetime Algebra....Pages 227-247
Laws of Reflection from Two or More Plane Mirrors in Succession....Pages 249-259
Exact Kinetic Energy Operators for Polyatomic Molecules....Pages 261-270
Geometry of Quantum Computing by Hamiltonian Dynamics of Spin Ensembles....Pages 271-283
Is the Brain a ‘Clifford Algebra Quantum Computer’?....Pages 285-295
A Hestenes Spacetime Algebra Approach to Light Polarization....Pages 297-306
Quaternions, Clifford Algebra and Symmetry Groups....Pages 307-315
Front Matter....Pages 317-317
A Generic Framework for Image Geometry....Pages 319-332
Color Edge Detection Using Rotors....Pages 333-339
Numerical Evaluation of Versors with Clifford Algebra....Pages 341-350
The Role of Clifford Algebra in Structure-Preserving Transformations for Second-Order Systems....Pages 351-359
Applications of Algebra of Incidence in Visually Guided Robotics....Pages 361-372
Monocular Pose Estimation of Kinematic Chains....Pages 373-383
Stabilization of 3D Pose Estimation....Pages 385-394
Inferring Dynamical Information from 3D Position Data using Geometric Algebra....Pages 395-406
Clifford Algebra Space Singularities of Inline Planar Platforms....Pages 407-421
Front Matter....Pages 423-423
Fast Quantum Fourier-Heisenberg-Weyl Transforms....Pages 425-435
The Structure Multivector....Pages 437-448
The Application of Clifford Algebra to Calculations of Multicomponent Chemical Composition....Pages 449-458
An Algorithm to Solve the Inverse IFS-Problem....Pages 459-467
Fast Quantum n-D Fourier and Radon Transforms....Pages 469-478
....