Visualizing Quaternions
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Visualizing Quaternions
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Author : Andrew J. Hanson
language : en
Publisher: Elsevier
Release Date : 2006-02-06
Visualizing Quaternions written by Andrew J. Hanson and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-02-06 with Mathematics categories.
Introduced 160 years ago as an attempt to generalize complex numbers to higher dimensions, quaternions are now recognized as one of the most important concepts in modern computer graphics. They offer a powerful way to represent rotations and compared to rotation matrices they use less memory, compose faster, and are naturally suited for efficient interpolation of rotations. Despite this, many practitioners have avoided quaternions because of the mathematics used to understand them, hoping that some day a more intuitive description will be available.The wait is over. Andrew Hanson's new book is a fresh perspective on quaternions. The first part of the book focuses on visualizing quaternions to provide the intuition necessary to use them, and includes many illustrative examples to motivate why they are important—a beautiful introduction to those wanting to explore quaternions unencumbered by their mathematical aspects. The second part covers the all-important advanced applications, including quaternion curves, surfaces, and volumes. Finally, for those wanting the full story of the mathematics behind quaternions, there is a gentle introduction to their four-dimensional nature and to Clifford Algebras, the all-encompassing framework for vectors and quaternions. - Richly illustrated introduction for the developer, scientist, engineer, or student in computer graphics, visualization, or entertainment computing. - Covers both non-mathematical and mathematical approaches to quaternions.
Visualizing More Quaternions
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Author : Andrew J. Hanson
language : en
Publisher: Elsevier
Release Date : 2024-07-01
Visualizing More Quaternions written by Andrew J. Hanson and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-07-01 with Mathematics categories.
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.
Dual Quaternions And Their Associated Clifford Algebras
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Author : Ronald Goldman
language : en
Publisher: CRC Press
Release Date : 2023-09-29
Dual Quaternions And Their Associated Clifford Algebras written by Ronald Goldman and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-09-29 with Computers categories.
Clifford algebra for dual quaternions has emerged recently as an alternative to standard matrix algebra as a computational framework for computer graphics. This book presents dual quaternions and their associated Clifford algebras in a new light, accessible to and geared toward the computer graphics community. Collecting all the associated formulas and theorems in one place, this book provides an extensive and rigorous treatment of dual quaternions, as well as showing how two models of Clifford algebra emerge naturally from the theory of dual quaternions. Each section comes complete with a set of exercises to help readers sharpen and practice their understanding. This book is accessible to anyone with a basic knowledge of quaternion algebra and is of particular use to forward-thinking members of the computer graphics community.
Rethinking Quaternions
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Author : Ron Goldman
language : en
Publisher: Springer Nature
Release Date : 2022-05-31
Rethinking Quaternions written by Ron Goldman and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-05-31 with Mathematics categories.
Quaternion multiplication can be used to rotate vectors in three-dimensions. Therefore, in computer graphics, quaternions have three principal applications: to increase speed and reduce storage for calculations involving rotations, to avoid distortions arising from numerical inaccuracies caused by floating point computations with rotations, and to interpolate between two rotations for key frame animation. Yet while the formal algebra of quaternions is well-known in the graphics community, the derivations of the formulas for this algebra and the geometric principles underlying this algebra are not well understood. The goals of this monograph are to provide a fresh, geometric interpretation for quaternions, appropriate for contemporary computer graphics, based on mass-points; to present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in three dimensions using insights from the algebra and geometry of multiplication in the complex plane; to derive the formula for quaternion multiplication from first principles; to develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection; to show how to apply sandwiching to compute perspective projections. In addition to these theoretical issues, we also address some computational questions. We develop straightforward formulas for converting back and forth between quaternion and matrix representations for rotations, reflections, and perspective projections, and we discuss the relative advantages and disadvantages of the quaternion and matrix representations for these transformations. Moreover, we show how to avoid distortions due to floating point computations with rotations by using unit quaternions to represent rotations. We also derive the formula for spherical linear interpolation, and we explain how to apply this formula to interpolate between two rotations for key frame animation. Finally, we explain the role of quaternions in low-dimensional Clifford algebras, and we show how to apply the Clifford algebra for R3 to model rotations, reflections, and perspective projections. To help the reader understand the concepts and formulas presented here, we have incorporated many exercises in order to clarify and elaborate some of the key points in the text. Table of Contents: Preface / Theory / Computation / Rethinking Quaternions and Clif ford Algebras / References / Further Reading / Author Biography
Quaternion Algebras
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Author : John Voight
language : en
Publisher: Springer Nature
Release Date : 2021-06-28
Quaternion Algebras written by John Voight and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-28 with Mathematics categories.
This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.
Intelligent Information And Database Systems
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Author : Ngoc-Thanh Nguyen
language : en
Publisher: Springer
Release Date : 2014-02-28
Intelligent Information And Database Systems written by Ngoc-Thanh Nguyen and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-02-28 with Computers categories.
The two-volume set LNAI 8397 and LNAI 8398 constitutes the refereed proceedings of the 6th Asian Conference on Intelligent Information and Database Systems, ACIIDS 2014, held in Bangkok, Thailand in April 2014. The 125 revised papers presented were carefully reviewed and selected from 300 submissions. Suggestion: The aim of the conference is to provide an internationally respected forum for scientific research in the technologies and applications of intelligent information and database systems. The papers are organized in topical sections on Natural Language and Text Processing, Intelligent Information Retrieval, Semantic Web, Social Networks and Recommendation Systems, Intelligent Database Systems, Decision Support Systems, Computer Vision Techniques, Machine Learning and Data Mining, Multiple Model Approach to Machine Learning, MMAML 2014, Computational Intelligence, CI 2014, Engineering Knowledge and Semantic Systems , IWEKSS 2014, Innovations in Intelligent Computation and Applications, IICA 2014, Modelling and Optimization Techniques in Information Systems, Database Systems and Industrial Systems, MOT 2014, Innovation via Collective Intelligences and Globalization in Business Management, ICIGBM 2014, Intelligent Supply Chains, ISC 2014, and Human Motion: Acquisition, Processing, Analysis, Synthesis and Visualization for Massive Datasets, HMMD 2014.
3d Math Primer For Graphics And Game Development
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Author : Fletcher Dunn
language : en
Publisher: CRC Press
Release Date : 2011-11-02
3d Math Primer For Graphics And Game Development written by Fletcher Dunn and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-11-02 with Computers categories.
This engaging book presents the essential mathematics needed to describe, simulate, and render a 3D world. Reflecting both academic and in-the-trenches practical experience, the authors teach you how to describe objects and their positions, orientations, and trajectories in 3D using mathematics. The text provides an introduction to mathematics for
Quaternion And Clifford Fourier Transforms And Wavelets
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Author : Eckhard Hitzer
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-24
Quaternion And Clifford Fourier Transforms And Wavelets written by Eckhard Hitzer and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-24 with Mathematics categories.
Quaternion and Clifford Fourier and wavelet transformations generalize the classical theory to higher dimensions and are becoming increasingly important in diverse areas of mathematics, physics, computer science and engineering. This edited volume presents the state of the art in these hypercomplex transformations. The Clifford algebras unify Hamilton’s quaternions with Grassmann algebra. A Clifford algebra is a complete algebra of a vector space and all its subspaces including the measurement of volumes and dihedral angles between any pair of subspaces. Quaternion and Clifford algebras permit the systematic generalization of many known concepts. This book provides comprehensive insights into current developments and applications including their performance and evaluation. Mathematically, it indicates where further investigation is required. For instance, attention is drawn to the matrix isomorphisms for hypercomplex algebras, which will help readers to see that software implementations are within our grasp. It also contributes to a growing unification of ideas and notation across the expanding field of hypercomplex transforms and wavelets. The first chapter provides a historical background and an overview of the relevant literature, and shows how the contributions that follow relate to each other and to prior work. The book will be a valuable resource for graduate students as well as for scientists and engineers.
Get Into Game Dev
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Author : Matthew Ventures
language : en
Publisher: CRC Press
Release Date : 2025-04-30
Get Into Game Dev written by Matthew Ventures and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-04-30 with Computers categories.
Get Into Game Dev: Tech Interview Tactics is a crash-course on how to pass a game development technical interview. It’s designed to guide intermediate and experienced coders through the depth and rigor necessary to land some of the most highly sought-after roles within interactive media. Unlike generic interview-prep books, GIGD maintains a laser-focus on game development to directly prepare candidates for roles like technical designer and gameplay engineer. Topics include 3D maths, programming fundamentals, and software design patterns. The author provides high quality instruction and practice problems based on his experience as a professional instructor and developer. Key Features: Includes an extensive set of practice questions taken from interviews of leading game development studios. Synthesizes coding and maths fundamentals into focused instruction, directly applicable to game development. Culminates in a rigorous practice test, designed to identify a reader’s weaknesses and guide them along the path to mastery. Uses a variety of mnemonics to assist readers in memorizing subject matter. Provides example worked solutions for readers to compare against their own problem-solving approaches. This book does not teach game development. Instead, it provides knowledge and instruction for a developer to achieve the technical mastery necessary to become a professional game developer.
Advanced Methods In Computer Graphics
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Author : Ramakrishnan Mukundan
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-02-10
Advanced Methods In Computer Graphics written by Ramakrishnan Mukundan and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-02-10 with Computers categories.
This book brings together several advanced topics in computer graphics that are important in the areas of game development, three-dimensional animation and real-time rendering. The book is designed for final-year undergraduate or first-year graduate students, who are already familiar with the basic concepts in computer graphics and programming. It aims to provide a good foundation of advanced methods such as skeletal animation, quaternions, mesh processing and collision detection. These and other methods covered in the book are fundamental to the development of algorithms used in commercial applications as well as research.