Visualizing More Quaternions - Hardcover

Hanson, Andrew J.

Hardcover

ISBN 10: 0323992021 ISBN 13: 9780323992022

Publisher: Morgan Kaufmann, 2024

Synopsis

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 Hanson’s own published and unpublished investigations relating to quaternion applications.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, and a detailed study of the Kleinian “ADE”
discrete groups of the ordinary two-sphere. Quaternion geometry is also incorporated into the isometric embedding of the Eguchi–Hanson gravitational instanton corresponding 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.

• 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.

About this title

About the Author

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.

From the Back Cover

Introduced 175 years ago as an attempt to generalize complex numbers to higher dimensions, quaternions are now recognized as one of the most important concepts in computer modelling, as they offer a powerful way to represent efficient interpolation of rotations. Visualizing More Quaternions is essentially a second volume of Dr. Andrew J. Hanson’s groundbreaking book, Visualizing Quaternions. This second volume updates and extends concepts that have evolved since the first book published in 2005 and adds entirely new insights that Dr. Hanson’s research has developed recently. This includes in particular the applications of quaternion methods to proteomics and molecular crystallography problems, which are domains with significant current research and application activity.The breadth of the topics to which Dr. Hanson has been applying quaternions has expanded significantly over the last decade, and Visualizing More Quaternions reflects the broader audience, in more diverse fields of science, involved with those scientific domains. In addition to readers interested in quaternions for their own sake, along with scientists involved in computer graphics, animation, shape modeling, and scientific visualization, 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. Dr. Hanson’s contribution to this domain is the formulation of protein amino acid signatures using per-residue quaternion-based frame triads, along with new methods for evaluating matching problems involving such frames.In addition to proteomics-related material useful for biochemists and biophysicists, Visualizing More Quaternions includes material that is related to electron microscopy (and specifically cryo-EM). The book also covers a spectrum of robotics problems that includes both orientation matching and alignment, and six-degree-of-freedom contexts based on dual quaternions, which were not covered at all in the author’s first book.Finally, there is a challenging set of chapters involving improved quaternion formulations of a wide range of mathematical physics problems that will appeal to readers involved in several domains of current research, including material related to gravitational instantons and Clifford algebras. Visualizing More Quaternions appeals to a broad population of scientific specialties, while treating a wide spectrum of difficulties, from fairly straightforward applications to very sophisticated applications. The book contains extensive computer graphics images of all sorts, including interactive device-independent web-based visualization applications that correspond to key static figures in the print version of the book, which are suitable for displaying the 4D properties of quaternions interactively.

Edition details

• Publisher: Morgan Kaufmann
• Publication date: 2024
• ISBN 10: 0323992021
• ISBN 13: 9780323992022
• Binding: Hardcover
• Language: English
• Edition number: 1
• Number of pages: 600