Visualizing More Quarternions
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Visualizing Quaternions
- Author : Andrew J. Hanson
- Publisher : Elsevier
- Release Date : 2006-02-06
- Total pages : 530
- ISBN : 9780080474779
- File Size : 48,5 Mb
- Total Download : 274
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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.
Rotations, Quaternions, and Double Groups
- Author : Simon L. Altmann
- Publisher : Courier Corporation
- Release Date : 2013-04-09
- Total pages : 336
- ISBN : 9780486317731
- File Size : 47,6 Mb
- Total Download : 810
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This self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to many illustrative problems. Geared toward upper-level undergraduates and graduate students, the book begins with chapters covering the fundamentals of symmetries, matrices, and groups, and it presents a primer on rotations and rotation matrices. Subsequent chapters explore rotations and angular momentum, tensor bases, the bilinear transformation, projective representations, and the geometry, topology, and algebra of rotations. Some familiarity with the basics of group theory is assumed, but the text assists students in developing the requisite mathematical tools as necessary.
Quaternions and Rotation Sequences
- Author : J. B. Kuipers
- Publisher : Princeton University Press
- Release Date : 2020-03-31
- Total pages : 229
- ISBN : 9780691211701
- File Size : 52,9 Mb
- Total Download : 459
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Ever since the Irish mathematician William Rowan Hamilton introduced quaternions in the nineteenth century--a feat he celebrated by carving the founding equations into a stone bridge--mathematicians and engineers have been fascinated by these mathematical objects. Today, they are used in applications as various as describing the geometry of spacetime, guiding the Space Shuttle, and developing computer applications in virtual reality. In this book, J. B. Kuipers introduces quaternions for scientists and engineers who have not encountered them before and shows how they can be used in a variety of practical situations. The book is primarily an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. But Kuipers also presents the more conventional and familiar 3 x 3 (9-element) matrix rotation operator. These parallel presentations allow the reader to judge which approaches are preferable for specific applications. The volume is divided into three main parts. The opening chapters present introductory material and establish the book's terminology and notation. The next part presents the mathematical properties of quaternions, including quaternion algebra and geometry. It includes more advanced special topics in spherical trigonometry, along with an introduction to quaternion calculus and perturbation theory, required in many situations involving dynamics and kinematics. In the final section, Kuipers discusses state-of-the-art applications. He presents a six degree-of-freedom electromagnetic position and orientation transducer and concludes by discussing the computer graphics necessary for the development of applications in virtual reality.
Quaternion Algebras
- Author : John Voight
- Publisher : Springer Nature
- Release Date : 2021-06-28
- Total pages : 885
- ISBN : 9783030566944
- File Size : 33,5 Mb
- Total Download : 727
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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.
Rethinking Quaternions
- Author : Ron Goldman
- Publisher : Morgan & Claypool Publishers
- Release Date : 2010
- Total pages : 157
- ISBN : 9781608454204
- File Size : 47,9 Mb
- Total Download : 242
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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."--Page 4 of cover.
3D Math Primer for Graphics and Game Development, 2nd Edition
- Author : Fletcher Dunn,Ian Parberry
- Publisher : CRC Press
- Release Date : 2011-11-02
- Total pages : 848
- ISBN : 9781568817231
- File Size : 53,8 Mb
- Total Download : 323
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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 game designers, including the fundamentals of coordinate spaces, vectors, and matrices. It also covers orientation in three dimensions, calculus and dynamics, graphics, and parametric curves.
Understanding Quaternions
- Author : Peng Du,Dong Ding ([Editor of Nova Science Publishers]),Zhuoyue Li ([Editor of Nova Science Publishers])
- Publisher : Nova Science Publishers
- Release Date : 2020
- Total pages : 197
- ISBN : 1536185531
- File Size : 40,5 Mb
- Total Download : 476
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"Quaternions are members of a noncommutative division algebra first invented by William Rowan Hamilton. They form an interesting algebra where each object contains 4 scalar variables, instead of Euler angles, which is useful to overcome the gimbal lock phenomenon when treating the rotation of objects. This book is about the mathematical basics and applications of quaternions. The first four chapters mainly concerns the mathematical theories, while the latter three chapters are related with three application aspects. It is expected to provide useful clues for researchers and engineers in the related area. In detail, this book is organized as follows: In Chapter 1, mathematical basics including the quaternion algebra and operations with quaternions, as well as the relationships of quaternions with other mathematical parameters and representations are demonstrated. In Chapter 2, how quaternions are formulated in Clifford Algebra, how it is used in explaining rotation group in symplectic vector space and parallel transformation in holonomic dynamics are presented. In Chapter 3, the wave equation for a spin 3/2 particle, described by 16-component vector-bispinor, is investigated in spherical coordinates. In Chapter 4, hyperbolic Lobachevsky and spherical Riemann models, parameterized coordinates with spherical and cylindric symmetry are studied. In Chapter 5, ship hydrodynamics with allowance of trim and sinkage is investigated and validated with experiments. In Chapter 6, the ballast flying phenomenon based on Discrete Discontinuous Analysis is presented. In Chapter 7, a numerical study is proposed to analyze the effect of the caisson sliding subjected to a hydrodynamic loading in the stability of the rear side of the rubble mound breakwater"--
Quaternion and Octonion Color Image Processing with MATLAB
- Author : Artyom M. Grigoryan,S. S. Agaian
- Publisher : Unknown
- Release Date : 2018
- Total pages : 404
- ISBN : 1510611363
- File Size : 46,8 Mb
- Total Download : 923
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"Color image processing has involved much interest in the recent years. The use of color in image processing is motivated by the facts that 1) the human eyes can discern thousands of colors, and image processing is used both for human interaction and computer interpretation; 2) the color image comprises more information than the gray-level image; 3) the color features are robust to several image processing procedures (for example, to the translation and rotation of the regions of interest); 4) the color features are efficiently used in many vision tasks, including object recognition and tracking, image segmentation and retrieval, image registration etc.; 5) the color is necessary in many real life applications such as visual communications, multimedia systems, fashion and food industries, computer vision, entertainment, consumer electronics, production printing and proofing, digital photography, biometrics, digital artwork reproduction, industrial inspection, and biomedical applications. Finally, the enormous number of color images that constantly are uploaded into Internet require new approaches and challenges of big visual media creation, retrieval, processing, and applications. It also gives us new opportunities to create a number of big visual data-driven applications. Three independent quantities are used to describe any particular color; the human eyes are seen all colors as variable combinations of primary colors of red, green, and blue. Many methods of the modern color image processing are based on dealing out each primary color"--
From Natural Numbers to Quaternions
- Author : Jürg Kramer,Anna-Maria von Pippich
- Publisher : Springer
- Release Date : 2017-11-15
- Total pages : 277
- ISBN : 9783319694290
- File Size : 28,8 Mb
- Total Download : 972
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This textbook offers an invitation to modern algebra through number systems of increasing complexity, beginning with the natural numbers and culminating with Hamilton's quaternions. Along the way, the authors carefully develop the necessary concepts and methods from abstract algebra: monoids, groups, rings, fields, and skew fields. Each chapter ends with an appendix discussing related topics from algebra and number theory, including recent developments reflecting the relevance of the material to current research. The present volume is intended for undergraduate courses in abstract algebra or elementary number theory. The inclusion of exercises with solutions also makes it suitable for self-study and accessible to anyone with an interest in modern algebra and number theory.
Graphics and Visualization
- Author : T. Theoharis,Georgios Papaioannou,Nikolaos Platis,Nicholas M. Patrikalakis
- Publisher : CRC Press
- Release Date : 2008-05-30
- Total pages : 786
- ISBN : 9781568812748
- File Size : 29,7 Mb
- Total Download : 984
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This book is a comprehensive introduction to visual computing, dealing with the modeling and synthesis of visual data by means of computers. What sets this book apart from other computer graphics texts is the integrated coverage of computer graphics and visualization topics, including important techniques such as subdivision and multi-resolution modeling, scene graphs, shadow generation, ambient occlusion, and scalar and vector data visualization. Students and practitioners will benefit from the comprehensive coverage of the principles that are the basic tools of their trade, from fundamental computer graphics and classic visualization techniques to advanced topics.
Intelligent Information and Database Systems
- Author : Ngoc-Thanh Nguyen,Boonwat Attachoo,Bogdan Trawinski,Kulwadee Somboonviwat
- Publisher : Springer
- Release Date : 2014-02-28
- Total pages : 618
- ISBN : 9783319054582
- File Size : 48,8 Mb
- Total Download : 107
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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.
Quaternions for Computer Graphics
- Author : John Vince
- Publisher : Springer Nature
- Release Date : 2021-09-02
- Total pages : 188
- ISBN : 9781447175094
- File Size : 12,9 Mb
- Total Download : 494
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If you have ever wondered what quaternions are — then look no further, John Vince will show you how simple and useful they are. This 2nd edition has been completely revised and includes extra detail on the invention of quaternions, a complete review of the text and equations, all figures are in colour, extra worked examples, an expanded index, and a bibliography arranged for each chapter. Quaternions for Computer Graphics includes chapters on number sets and algebra, imaginary and complex numbers, the complex plane, rotation transforms, and a comprehensive description of quaternions in the context of rotation. The book will appeal to students of computer graphics, computer science and mathematics, as well as programmers, researchers, academics and professional practitioners interested in learning about quaternions. John Vince explains in an easy-to-understand language, with the aid of useful figures, how quaternions emerged, gave birth to modern vector analysis, disappeared, and reemerged to be adopted by the flight simulation industry and computer graphics. This book will give you the confidence to use quaternions within your every-day mathematics, and explore more advanced texts.
Real Quaternionic Calculus Handbook
- Author : João Pedro Morais,Svetlin Georgiev,Wolfgang Sprößig
- Publisher : Springer Science & Business Media
- Release Date : 2014-01-08
- Total pages : 216
- ISBN : 9783034806220
- File Size : 41,7 Mb
- Total Download : 788
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Real quaternion analysis is a multi-faceted subject. Created to describe phenomena in special relativity, electrodynamics, spin etc., it has developed into a body of material that interacts with many branches of mathematics, such as complex analysis, harmonic analysis, differential geometry, and differential equations. It is also a ubiquitous factor in the description and elucidation of problems in mathematical physics. In the meantime real quaternion analysis has become a well established branch in mathematics and has been greatly successful in many different directions. This book is based on concrete examples and exercises rather than general theorems, thus making it suitable for an introductory one- or two-semester undergraduate course on some of the major aspects of real quaternion analysis in exercises. Alternatively, it may be used for beginning graduate level courses and as a reference work. With exercises at the end of each chapter and its straightforward writing style the book addresses readers who have no prior knowledge on this subject but have a basic background in graduate mathematics courses, such as real and complex analysis, ordinary differential equations, partial differential equations, and theory of distributions.
Visual Complex Analysis
- Author : Tristan Needham
- Publisher : Oxford University Press
- Release Date : 1997
- Total pages : 620
- ISBN : 0198534469
- File Size : 28,6 Mb
- Total Download : 514
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Now available in paperback, this successful radical approach to complex analysis replaces the standard calculational arguments with new geometric ones. With several hundred diagrams, and far fewer prerequisites than usual, this is the first visual intuitive introduction to complex analysis.Although designed for use by undergraduates in mathematics and science, the novelty of the approach will also interest professional mathematicians.
Computer Visualization
- Author : Richard S. Gallagher,0 Solomon Press,,
- Publisher : CRC Press
- Release Date : 2023-05-31
- Total pages : 328
- ISBN : 9781000939811
- File Size : 36,8 Mb
- Total Download : 612
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Rapid advances in 3-D scientific visualization have made a major impact on the display of behavior. The use of 3-D has become a key component of both academic research and commercial product development in the field of engineering design. Computer Visualization presents a unified collection of computer graphics techniques for the scientific visualization of behavior. The book combines a basic overview of the fundamentals of computer graphics with a practitioner-oriented review of the latest 3-D graphics display and visualization techniques. Each chapter is written by well-known experts in the field. The first section reviews how computer graphics visualization techniques have evolved to work with digital numerical analysis methods. The fundamentals of computer graphics that apply to the visualization of analysis data are also introduced. The second section presents a detailed discussion of the algorithms and techniques used to visualize behavior in 3-D, as static, interactive, or animated imagery. It discusses the mathematics of engineering data for visualization, as well as providing the current methods used for the display of scalar, vector, and tensor fields. It also examines the more general issues of visualizing a continuum volume field and animating the dimensions of time and motion in a state of behavior. The final section focuses on production visualization capabilities, including the practical computational aspects of visualization such as user interfaces, database architecture, and interaction with a model. The book concludes with an outline of successful practical applications of visualization, and future trends in scientific visualization.
Advances in Robot Kinematics 2018
- Author : Jadran Lenarcic,Vincenzo Parenti-Castelli
- Publisher : Springer
- Release Date : 2018-06-22
- Total pages : 477
- ISBN : 9783319931883
- File Size : 16,7 Mb
- Total Download : 351
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This is the proceedings of ARK 2018, the 16th International Symposium on Advances in Robot Kinematics, that was organized by the Group of Robotics, Automation and Biomechanics (GRAB) from the University of Bologna, Italy. ARK are international symposia of the highest level organized every two years since 1988. ARK provides a forum for researchers working in robot kinematics and stimulates new directions of research by forging links between robot kinematics and other areas.The main topics of the symposium of 2018 were: kinematic analysis of robots, robot modeling and simulation, kinematic design of robots, kinematics in robot control, theories and methods in kinematics, singularity analysis, kinematic problems in parallel robots, redundant robots, cable robots, over-constrained linkages, kinematics in biological systems, humanoid robots and humanoid subsystems.
Real-Time Rendering
- Author : Tomas Akenine-Möller,Eric Haines,Naty Hoffman
- Publisher : CRC Press
- Release Date : 2019-01-18
- Total pages : 1046
- ISBN : 9781315362007
- File Size : 54,7 Mb
- Total Download : 501
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Thoroughly revised, this third edition focuses on modern techniques used to generate synthetic three-dimensional images in a fraction of a second. With the advent of programmable shaders, a wide variety of new algorithms have arisen and evolved over the past few years. This edition discusses current, practical rendering methods used in games and other applications. It also presents a solid theoretical framework and relevant mathematics for the field of interactive computer graphics, all in an approachable style. The authors have made the figures used in the book available for download for fair use.:Download Figures. Reviews Rendering has been a required reference for professional graphics practitioners for nearly a decade. This latest edition is as relevant as ever, covering topics from essential mathematical foundations to advanced techniques used by today’s cutting edge games. -- Gabe Newell, President, Valve, May 2008 Rendering ... has been completely revised and revamped for its updated third edition, which focuses on modern techniques used to generate three-dimensional images in a fraction of the time old processes took. From practical rendering for games to math and details for better interactive applications, it's not to be missed. -- The Bookwatch, November 2008 You'll get brilliantly lucid explanations of concepts like vertex morphing and variance shadow mapping—as well as a new respect for the incredible craftsmanship that goes into today's PC games. -- Logan Decker, PC Gamer Magazine , February 2009
Geometric Algebra Applications Vol. I
- Author : Eduardo Bayro-Corrochano
- Publisher : Springer
- Release Date : 2018-06-20
- Total pages : 742
- ISBN : 9783319748306
- File Size : 25,5 Mb
- Total Download : 199
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The goal of the Volume I Geometric Algebra for Computer Vision, Graphics and Neural Computing is to present a unified mathematical treatment of diverse problems in the general domain of artificial intelligence and associated fields using Clifford, or geometric, algebra. Geometric algebra provides a rich and general mathematical framework for Geometric Cybernetics in order to develop solutions, concepts and computer algorithms without losing geometric insight of the problem in question. Current mathematical subjects can be treated in an unified manner without abandoning the mathematical system of geometric algebra for instance: multilinear algebra, projective and affine geometry, calculus on manifolds, Riemann geometry, the representation of Lie algebras and Lie groups using bivector algebras and conformal geometry. By treating a wide spectrum of problems in a common language, this Volume I offers both new insights and new solutions that should be useful to scientists, and engineers working in different areas related with the development and building of intelligent machines. Each chapter is written in accessible terms accompanied by numerous examples, figures and a complementary appendix on Clifford algebras, all to clarify the theory and the crucial aspects of the application of geometric algebra to problems in graphics engineering, image processing, pattern recognition, computer vision, machine learning, neural computing and cognitive systems.
Chaos and Fractals
- Author : C.A. Pickover
- Publisher : Elsevier
- Release Date : 1998-08-03
- Total pages : 452
- ISBN : 0080528864
- File Size : 11,7 Mb
- Total Download : 712
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These days computer-generated fractal patterns are everywhere, from squiggly designs on computer art posters to illustrations in the most serious of physics journals. Interest continues to grow among scientists and, rather surprisingly, artists and designers. This book provides visual demonstrations of complicated and beautiful structures that can arise in systems, based on simple rules. It also presents papers on seemingly paradoxical combinations of randomness and structure in systems of mathematical, physical, biological, electrical, chemical, and artistic interest. Topics include: iteration, cellular automata, bifurcation maps, fractals, dynamical systems, patterns of nature created through simple rules, and aesthetic graphics drawn from the universe of mathematics and art. Chaos and Fractals is divided into six parts: Geometry and Nature; Attractors; Cellular Automata, Gaskets, and Koch Curves; Mandelbrot, Julia and Other Complex Maps; Iterated Function Systems; and Computer Art. Additionally, information on the latest practical applications of fractals and on the use of fractals in commercial products such as the antennas and reaction vessels is presented. In short, fractals are increasingly finding application in practical products where computer graphics and simulations are integral to the design process. Each of the six sections has an introduction by the editor including the latest research, references, and updates in the field. This book is enhanced with numerous color illustrations, a comprehensive index, and the many computer program examples encourage reader involvement.
A Concise Introduction to Scientific Visualization
- Author : Brad Eric Hollister,Alex Pang
- Publisher : Springer Nature
- Release Date : 2022-01-01
- Total pages : 107
- ISBN : 9783030864194
- File Size : 18,6 Mb
- Total Download : 644
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Scientific visualization has always been an integral part of discovery, starting first with simplified drawings of the pre-Enlightenment and progressing to present day. Mathematical formalism often supersedes visual methods, but their use is at the core of the mental process. As historical examples, a spatial description of flow led to electromagnetic theory, and without visualization of crystals, structural chemistry would not exist. With the advent of computer graphics technology, visualization has become a driving force in modern computing. A Concise Introduction to Scientific Visualization – Past, Present, and Future serves as a primer to visualization without assuming prior knowledge. It discusses both the history of visualization in scientific endeavour, and how scientific visualization is currently shaping the progress of science as a multi-disciplinary domain.
Mathematics for Computer Graphics
- Author : John Vince
- Publisher : Springer Science & Business Media
- Release Date : 2005-12-27
- Total pages : 248
- ISBN : 9781846282836
- File Size : 43,8 Mb
- Total Download : 545
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This is a concise and informal introductory book on the mathematical concepts that underpin computer graphics. The author, John Vince, makes the concepts easy to understand, enabling non-experts to come to terms with computer animation work. The book complements the author's other works and is written in the same accessible and easy-to-read style. It is also a useful reference book for programmers working in the field of computer graphics, virtual reality, computer animation, as well as students on digital media courses, and even mathematics courses.