Visualizing More Quaternions

Andrew J. Hanson

• Publisher: Morgan Kaufmann
• Year: 2024
• ISBN:
  9780323993906
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Visualizing More Quaternions, Volume Two updates on proteomics-related material that will be useful for biochemists and biophysicists, including material related to electron microscopy (and specifically cryo-EVisualizing. Dr. Andrew J. Hanson's groundbreaking book updates and extends concepts that have evolved since the first book published in 2005, adding entirely new insights that Dr. Hanson's research has recently developed. This includes the applications of quaternion methods to proteomics and molecular crystallography problems, which are domains with significant current research and application activity.In addition to readers interested in quaternions for their own sake, scientists involved in computer graphics, animation, shape modeling, and scientific visualization, and readers from several other disciplines will benefit from this new volume. Foremost among these, and the target of the first several chapters, are scientists involved in molecular chemistry where techniques based on quaternion eigensystems have become a standard tool for evaluating the quality of shape matching.

• Establishes basic principles for visual display of quaternions and their applications.

• Explores quaternion based approaches to the matching of point cloud pairs, including approaches to data from orthographic and perspective projections.
  
  

• Develops extensive applications of quaternion frames to protein orientation analysis.
  
  

• Analyzes the application of quaternion methods to physics problems ranging from quantum computing to special relativity and gravitational instantons.
  
  

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Dedication
• Preface
• Biography
• Part 1: Review
• Introduction
• Chapter 1: Introduction
• Chapter 2: Basic quaternion formulas
• Chapter 3: What are quaternions?
• Chapter 4: What is quaternion visualization?
• Chapter 5: Interacting with quaternions as hyperspheres
• Chapter 6: Quaternions and uniform rotation distributions
• Part 2: Orientation matching
• Introduction
• Chapter 7: Introduction: 2D cloud alignment problem
• Chapter 8: The 3D quaternion-based alignment problem
• Chapter 9: Quaternion of a 3D rotation from the Bar-Itzhack method
• Chapter 10: The 3D quaternion-based frame alignment problem
• Chapter 11: The combined point-frame alignment problem
• Part 3: Quaternion adjugate methods for pose estimation
• Introduction
• Chapter 12: Exploring the quaternion adjugate variables
• Chapter 13: Quaternion-related machine learning
• Chapter 14: Exact 2D rotations for cloud matching and pose estimation tasks
• Chapter 15: Exact 3D rotations for cloud matching and pose estimation tasks
• Part 4: Quaternion proteomics
• Introduction
• Chapter 16: Quaternion protein frame maps
• Part 5: Spherical geometry and quaternions
• Introduction
• Chapter 17: Euclidean center of mass and barycentric coordinates
• Chapter 18: Quaternion averaging and the barycenter
• Chapter 19: The quaternion barycentric coordinate problem
• Part 6: Extending quaternions to 4D and SE(3)
• Introduction
• Chapter 20: The 4D quaternion-based coordinate and orientation frame alignment problems
• Chapter 21: Dual quaternions and SE(3)
• Part 7: Quaternion applications in physics
• Introduction
• Chapter 22: Quantum computing and quaternion qubits
• Chapter 23: Introduction to quaternions and special relativity
• Chapter 24: Complex quaternions and special relativity
• Part 8: Quaternion discrete symmetries and gravitons
• Introduction
• Chapter 25: Geometry of the ADE symmetries of the 2-sphere
• Chapter 26: The discrete ADE rotation groups and their quaternions
• Chapter 27: The quaternion presentation form of the discrete ADE groups
• Chapter 28: The Klein invariants of the ADE discrete quaternion groups
• Chapter 29: Quaternion-based embedding and visualization of the A1 gravitational instanton
• Chapter 30: Conclusion
• Appendix A: Notation
• Appendix B: Quaternion methods
• Appendix C: Eigensystem of the rotation matrix
• Appendix D: Lie algebra origins of the rotation matrix
• Appendix E: Derivation of rotation group representations
• Appendix F: Quaternion from double 3D Clifford reflection
• Appendix G: Example code for rotation to quaternion algorithm
• Appendix H: Selected details of the 3D perspective solution
• Appendix I: Exact cloud matching rotations in any dimension
• Appendix J: Details of the Cardano fourth-order eigenvalue solution
• Appendix K: A quaternion context for traditional Ramachandran protein plots
• Appendix L: Determinants and the Levi-Civita symbol
• Appendix M: The dual basis
• Appendix N: Equivalence of SVD and quaternion eigensystem methods
• References
• Index