Understanding Quaternions

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Understanding Quaternions

Peng Du,Dong Ding ([Editor of Nova Science Publishers]),Zhuoyue Li ([Editor of Nova Science Publishers])
Understanding Quaternions Book PDF

Author : Peng Du,Dong Ding ([Editor of Nova Science Publishers]),Zhuoyue Li ([Editor of Nova Science Publishers])
Publisher : Nova Science Publishers
Page : 197 pages
File Size : 46,9 Mb
Release : 2020
Category : Mathematics
ISBN : 1536185531

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Understanding Quaternions by Peng Du,Dong Ding ([Editor of Nova Science Publishers]),Zhuoyue Li ([Editor of Nova Science Publishers]) Pdf

"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"--

Quaternions for Computer Graphics

John Vince
Quaternions for Computer Graphics Book PDF

Author : John Vince
Publisher : Springer Nature
Page : 188 pages
File Size : 48,6 Mb
Release : 2021-09-02
Category : Computers
ISBN : 9781447175094

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Quaternions for Computer Graphics by John Vince Pdf

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.

Rotations, Quaternions, and Double Groups

Simon L. Altmann
Rotations Quaternions and Double Groups Book PDF

Author : Simon L. Altmann
Publisher : Courier Corporation
Page : 315 pages
File Size : 43,6 Mb
Release : 2013-04-09
Category : Mathematics
ISBN : 9780486317731

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Rotations, Quaternions, and Double Groups by Simon L. Altmann Pdf

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.

Rethinking Quaternions

Ron Goldman
Rethinking Quaternions Book PDF

Author : Ron Goldman
Publisher : Springer Nature
Page : 157 pages
File Size : 49,7 Mb
Release : 2022-05-31
Category : Mathematics
ISBN : 9783031795497

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Rethinking Quaternions by Ron Goldman Pdf

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

Hands-On C++ Game Animation Programming

Gabor Szauer
Hands On C Game Animation Programming Book PDF

Author : Gabor Szauer
Publisher : Packt Publishing Ltd
Page : 368 pages
File Size : 40,5 Mb
Release : 2020-06-12
Category : Computers
ISBN : 9781800207967

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Hands-On C++ Game Animation Programming by Gabor Szauer Pdf

Learn animation programming from first principles and implement modern animation techniques that can be integrated into any game development workflow Key FeaturesBuild a functional and production-ready modern animation system with complete features using C++Learn basic, advanced, and skinned animation programming with this step-by-step guideDiscover the math required to implement cutting edge animation techniques such as inverse kinematics and dual quaternionsBook Description Animation is one of the most important parts of any game. Modern animation systems work directly with track-driven animation and provide support for advanced techniques such as inverse kinematics (IK), blend trees, and dual quaternion skinning. This book will walk you through everything you need to get an optimized, production-ready animation system up and running, and contains all the code required to build the animation system. You’ll start by learning the basic principles, and then delve into the core topics of animation programming by building a curve-based skinned animation system. You’ll implement different skinning techniques and explore advanced animation topics such as IK, animation blending, dual quaternion skinning, and crowd rendering. The animation system you will build following this book can be easily integrated into your next game development project. The book is intended to be read from start to finish, although each chapter is self-contained and can be read independently as well. By the end of this book, you’ll have implemented a modern animation system and got to grips with optimization concepts and advanced animation techniques. What you will learnGet the hang of 3D vectors, matrices, and transforms, and their use in game developmentDiscover various techniques to smoothly blend animationsGet to grips with GLTF file format and its design decisions and data structuresDesign an animation system by using animation tracks and implementing skinningOptimize various aspects of animation systems such as skinned meshes, clip sampling, and pose palettesImplement the IK technique for your game characters using CCD and FABRIK solversUnderstand dual quaternion skinning and how to render large instanced crowdsWho this book is for This book is for professional, independent, and hobbyist developers interested in building a robust animation system from the ground up. Some knowledge of the C++ programming language will be helpful.

Visualizing Quaternions

Andrew J. Hanson
Visualizing Quaternions Book PDF

Author : Andrew J. Hanson
Publisher : Elsevier
Page : 530 pages
File Size : 42,6 Mb
Release : 2006-02-06
Category : Mathematics
ISBN : 9780080474779

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Visualizing Quaternions by Andrew J. Hanson Pdf

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.

Quaternions and Rotation Sequences

J. B. Kuipers
Quaternions and Rotation Sequences Book PDF

Author : J. B. Kuipers
Publisher : Princeton University Press
Page : 396 pages
File Size : 43,8 Mb
Release : 2020-03-31
Category : Mathematics
ISBN : 9780691211701

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Quaternions and Rotation Sequences by J. B. Kuipers Pdf

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

John Voight
Quaternion Algebras Book PDF

Author : John Voight
Publisher : Springer Nature
Page : 877 pages
File Size : 48,6 Mb
Release : 2021-06-28
Category : Mathematics
ISBN : 9783030566944

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Quaternion Algebras by John Voight Pdf

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.

Dual Quaternions and Their Associated Clifford Algebras

Ronald Goldman
Dual Quaternions and Their Associated Clifford Algebras Book PDF

Author : Ronald Goldman
Publisher : CRC Press
Page : 215 pages
File Size : 54,5 Mb
Release : 2023-09-29
Category : Computers
ISBN : 9781000903522

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Dual Quaternions and Their Associated Clifford Algebras by Ronald Goldman Pdf

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.

Rigid Body Dynamics for Beginners

Phil Kim
Rigid Body Dynamics for Beginners Book PDF

Author : Phil Kim
Publisher : Createspace Independent Publishing Platform
Page : 0 pages
File Size : 55,5 Mb
Release : 2013
Category : Dynamics, Rigid
ISBN : 1493598201

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Rigid Body Dynamics for Beginners by Phil Kim Pdf

Has anyone experienced difficulty and confusion in understanding what the Euler angles, quaternions, and direction cosine matrices are, and furthermore, about the relationships among them? This book is for those who had struggled to figure out what all the aforementioned concepts are, and also provides a practical example that could be easily followed with MATLAB. In addition, some surface of linear motions will also be touched so that in the end, a set of equations of motion that describe the motion of a rigid body in three-dimensional space could be constructed. Following step by step, the reader will be gradually immersed into the joy of learning and applying basic attitude dynamics. The book will be good for those who are already familiar in the field by helping them reorganize the concepts and knowledge they have learned before.

Quaternion Cosmos

Wardell Lindsay
Quaternion Cosmos Book PDF

Author : Wardell Lindsay
Publisher : Lulu.com
Page : 57 pages
File Size : 52,9 Mb
Release : 2006-07-01
Category : Technology & Engineering
ISBN : 9781847281395

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Quaternion Cosmos by Wardell Lindsay Pdf

Truth is Beauty and Beauty is Truth. A Four dimensional Theory of Physics for the Four Dimensional Cosmos. Physics for the engineers of the future.

Unreal Engine 4.x Scripting with C++ Cookbook

John P. Doran,William Sherif,Stephen Whittle
Unreal Engine 4 x Scripting with C Cookbook Book PDF

Author : John P. Doran,William Sherif,Stephen Whittle
Publisher : Packt Publishing Ltd
Page : 693 pages
File Size : 47,9 Mb
Release : 2019-03-29
Category : Computers
ISBN : 9781789803372

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Unreal Engine 4.x Scripting with C++ Cookbook by John P. Doran,William Sherif,Stephen Whittle Pdf

Unreal Engine 4 (UE4) is a popular and award-winning game engine that powers some of the most popular games. A truly powerful tool for game development, there has never been a better time to use it for both commercial and independent projects. With more than 100 recipes, this book shows how to unleash the power of C++ while developing games ...

Advances in Imaging and Electron Physics

Anonim
Advances in Imaging and Electron Physics Book PDF

Author : Anonim
Publisher : Academic Press
Page : 361 pages
File Size : 46,5 Mb
Release : 2013-01-17
Category : Technology & Engineering
ISBN : 9780124077973

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Advances in Imaging and Electron Physics by Anonim Pdf

Advances in Imaging and Electron Physics features cutting-edge articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science and digital image processing, electromagnetic wave propagation, electron microscopy, and the computing methods used in all these domains. Contributions from leading authorities Informs and updates on all the latest developments in the field

Understanding the Discrete Element Method

Hans-Georg Matuttis,Jian Chen
Understanding the Discrete Element Method Book PDF

Author : Hans-Georg Matuttis,Jian Chen
Publisher : John Wiley & Sons
Page : 484 pages
File Size : 52,8 Mb
Release : 2014-05-12
Category : Science
ISBN : 9781118567289

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Understanding the Discrete Element Method by Hans-Georg Matuttis,Jian Chen Pdf

Gives readers a more thorough understanding of DEM and equips researchers for independent work and an ability to judge methods related to simulation of polygonal particles Introduces DEM from the fundamental concepts (theoretical mechanics and solidstate physics), with 2D and 3D simulation methods for polygonal particles Provides the fundamentals of coding discrete element method (DEM) requiring little advance knowledge of granular matter or numerical simulation Highlights the numerical tricks and pitfalls that are usually only realized after years of experience, with relevant simple experiments as applications Presents a logical approach starting withthe mechanical and physical bases,followed by a description of the techniques and finally their applications Written by a key author presenting ideas on how to model the dynamics of angular particles using polygons and polyhedral Accompanying website includes MATLAB-Programs providing the simulation code for two-dimensional polygons Recommended for researchers and graduate students who deal with particle models in areas such as fluid dynamics, multi-body engineering, finite-element methods, the geosciences, and multi-scale physics.

Conformal Geometry of Surfaces in S4 and Quaternions

Francis E. Burstall
Conformal Geometry of Surfaces in S4 and Quaternions Book PDF

Author : Francis E. Burstall
Publisher : Springer Science & Business Media
Page : 104 pages
File Size : 49,8 Mb
Release : 2002-03-05
Category : Mathematics
ISBN : 3540430083

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Conformal Geometry of Surfaces in S4 and Quaternions by Francis E. Burstall Pdf

The conformal geometry of surfaces recently developed by the authors leads to a unified understanding of algebraic curve theory and the geometry of surfaces on the basis of a quaternionic-valued function theory. The book offers an elementary introduction to the subject but takes the reader to rather advanced topics. Willmore surfaces in the foursphere, their Bcklund and Darboux transforms are covered, and a new proof of the classification of Willmore spheres is given.