Visualizing More Quaternions

Visualizing More Quaternions is a sequel to Dr. Andrew J. Hanson?s first book, Visualizing Quaternions, which appeared in 2006. This new volume develops and extends concepts that have attracted the author?s attention in the intervening 18 years, providing new insights into existing scholarship, and detailing results from Dr. Hanson?s own published and unpublished investigations relating to quaternion applications. Among the topics covered are the introduction of new approaches to depicting quaternions and their properties, applications of quaternion methods to cloud matching, including both orthographic and perspective projection problems, and orientation feature analysis for proteomics and bioinformatics. The quaternion adjugate variables are introduced to embody the nontrivial quaternion topology on the three-sphere and incorporate it into machine learning tasks. Other subjects include quaternion applications to a wide variety of problems in physics, including quantum computing, complexified quaternions in special relativity, a detailed study of the Kleinian ?ADE” discrete groups of the ordinary two-sphere, and the incorporation of quaternion geometry into the isometric embedding of the Eguchi-Hanson gravitational instanton that corresponds to the k=1 Kleinian cyclic group. Visualizing More Quaternions endeavors to explore novel ways of thinking about challenging problems that are relevant to a broad audience involved in a wide variety of scientific disciplines.

1. Introduction and Review of Quaternion Methodology
2. Synopsis of Useful Applications of Quaternions
3. Quaternion maps of global protein orientation-frame structure
4. Quaternion methods for the Root Mean Square Deviation (RMSD) problem
5. Quaternion methods for the Quaternion Frame Alignment (QFA) problem for 3D molecular and protein structures
6. Quaternion methods for the RMSD coordinate matching problem in 4D
7. Quaternion methods for Quaternion Frame Alignment QFA of molecular and protein structures in 4D
8. Quaternion methods for the robotics hand-eye matching problem
9. Quaternion methods for the machine vision multicamera alignment problem
10. Elements and applications of Dual Quaternions for 6 degree-of-freedom motion analysis
11. Bar-Itzhack method for finding quaternions from numerical rotation matrices
12. Methods for calculating the quaternion barycenter
13. Quaternion frame approach to Cryo-Electron-Microscopy
14. Extraction of quaternion camera viewpoint information from 3D data using deep AI and machine learning methods
15. Quaternion projective geometry and invariant cross-ratios
16. Special Relativity using complexified quaternions18. Relativistic spinning top dynamics with quaternions
17. Skyrmions in Quaternion Form
18. Quaternion maps of the Kleinian Groups
19. Extension of quaternion shape matching methods in 3D and 4D to arbitrary dimension N using Clifford Algebras
20. Conclusion and Philosophical Remarks on Quaternions

Andrew J. Hanson Ph.D. is an Emeritus Professor of Computer Science at Indiana University. He earned a bachelor’s degree in Chemistry and Physics from Harvard University in 1966 and a PhD in Theoretical Physics from MIT under Kerson Huang in 1971. His interests range from general relativity to computer graphics, artificial intelligence, and bioinformatics; he is particularly concerned with applications of quaternions and with exploitation of higher-dimensional graphics for the visualization of complex scientific contexts such as Calabi-Yau spaces. He is the co-discoverer of the Eguchi-Hanson “gravitational instanton” Einstein metric (1978), author of Visualizing Quaternions (Elsevier, 2006), and designer of the iPhone Apps “4Dice” and “4DRoom” (2012) for interacting with four-dimensional virtual reality.