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Quaternions for Computer Graphics

Quaternions for Computer Graphics Author John Vince
ISBN-10 0857297600
Release 2011-06-11
Pages 140
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Sir William Rowan Hamilton was a genius, and will be remembered for his significant contributions to physics and mathematics. The Hamiltonian, which is used in quantum physics to describe the total energy of a system, would have been a major achievement for anyone, but Hamilton also invented quaternions, which paved the way for modern vector analysis. Quaternions are one of the most documented inventions in the history of mathematics, and this book is about their invention, and how they are used to rotate vectors about an arbitrary axis. Apart from introducing the reader to the features of quaternions and their associated algebra, the book provides valuable historical facts that bring the subject alive. Quaternions for Computer Graphics introduces the reader to quaternion algebra by describing concepts of sets, groups, fields and rings. It also includes chapters on imaginary quantities, complex numbers and the complex plane, which are essential to understanding quaternions. The book contains many illustrations and worked examples, which make it essential reading for students, academics, researchers and professional practitioners.



Quaternions for Computer Graphics

Quaternions for Computer Graphics Author John Vince
ISBN-10 0857297619
Release 2011-06-18
Pages 140
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Sir William Rowan Hamilton was a genius, and will be remembered for his significant contributions to physics and mathematics. The Hamiltonian, which is used in quantum physics to describe the total energy of a system, would have been a major achievement for anyone, but Hamilton also invented quaternions, which paved the way for modern vector analysis. Quaternions are one of the most documented inventions in the history of mathematics, and this book is about their invention, and how they are used to rotate vectors about an arbitrary axis. Apart from introducing the reader to the features of quaternions and their associated algebra, the book provides valuable historical facts that bring the subject alive. Quaternions for Computer Graphics introduces the reader to quaternion algebra by describing concepts of sets, groups, fields and rings. It also includes chapters on imaginary quantities, complex numbers and the complex plane, which are essential to understanding quaternions. The book contains many illustrations and worked examples, which make it essential reading for students, academics, researchers and professional practitioners.



Visualizing Quaternions

Visualizing Quaternions Author Andrew J. Hanson
ISBN-10 9780080474779
Release 2006-02-06
Pages 530
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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.



Rethinking Quaternions

Rethinking Quaternions Author Ron Goldman
ISBN-10 9781608454204
Release 2010
Pages 157
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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.



Mathematics for Computer Graphics

Mathematics for Computer Graphics Author John Vince
ISBN-10 9781447173366
Release 2017-08-28
Pages 505
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John Vince explains a wide range of mathematical techniques and problem-solving strategies associated with computer games, computer animation, virtual reality, CAD and other areas of computer graphics in this completely revised and expanded fifth edition. The first five chapters cover a general introduction, number sets, algebra, trigonometry and coordinate systems, which are employed in the following chapters on vectors, matrix algebra, transforms, interpolation, curves and patches, analytic geometry and barycentric coordinates. Following this, the reader is introduced to the relatively new topic of geometric algebra, followed by two chapters that introduce differential and integral calculus. Finally, there is a chapter on worked examples. Mathematics for Computer Graphics covers all of the key areas of the subject, including: · Number sets · Algebra · Trigonometry · Coordinate systems · Determinants · Vectors · Quaternions · Matrix algebra · Geometric transforms · Interpolation · Curves and surfaces · Analytic geometry · Barycentric coordinates · Geometric algebra · Differential calculus · Integral calculus This fifth edition contains over 120 worked examples and over 320 colour illustrations, which are central to the author’s descriptive writing style. Mathematics for Computer Graphics provides a sound understanding of the mathematics required for computer graphics, giving a fascinating insight into the design of computer graphics software and setting the scene for further reading of more advanced books and technical research papers.



Rotation Transforms for Computer Graphics

Rotation Transforms for Computer Graphics Author John Vince
ISBN-10 0857291548
Release 2011-01-04
Pages 232
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Rotation transforms are used everywhere in computer graphics from rotating pictures in editing software, to providing an arbitrary view of a 3D virtual environment. Although the former is a trivial operation, the latter can be a challenging task. Rotation Transforms for Computer Graphics covers a wide range of mathematical techniques used for rotating points and frames of reference in the plane and 3D space. It includes many worked examples and over 100 illustrations that make it essential reading for students, academics, researchers and professional practitioners. The book includes introductory chapters on complex numbers, matrices, quaternions and geometric algebra, and further chapters on how these techniques are employed in 2D and 3D computer graphics. In particular, matrix and bivector transforms are developed and evaluated to rotate points in a fixed frame of reference, and vice versa.



Quaternions and Rotation Sequences

Quaternions and Rotation Sequences Author Jack B. Kuipers
ISBN-10 0691102988
Release 2002
Pages 371
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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.



Geometry for Computer Graphics

Geometry for Computer Graphics Author John Vince
ISBN-10 1852338342
Release 2005-01-05
Pages 342
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This book draws together a wide variety of geometric information that will provide a sourcebook of facts, examples and proofs for students, academics, researchers and professional practitioners. It includes a summary hundreds of formulae used to solve 2D and 3D geometric problems; worked examples; proofs; mathematical strategies for solving geometric problems; a glossary of terms used in geometry.



Applied Geometry for Computer Graphics and CAD

Applied Geometry for Computer Graphics and CAD Author Duncan Marsh
ISBN-10 9781846281099
Release 2006-03-30
Pages 350
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Focusing on the manipulation and representation of geometrical objects, this book explores the application of geometry to computer graphics and computer-aided design (CAD). Over 300 exercises are included, some new to this edition, and many of which encourage the reader to implement the techniques and algorithms discussed through the use of a computer package with graphing and computer algebra capabilities. A dedicated website also offers further resources and useful links.



Curves and Surfaces for Computer Graphics

Curves and Surfaces for Computer Graphics Author David Salomon
ISBN-10 9780387284521
Release 2007-03-20
Pages 460
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Requires only a basic knowledge of mathematics and is geared toward the general educated specialists. Includes a gallery of color images and Mathematica code listings.



Quaternions Spinors and Surfaces

Quaternions  Spinors  and Surfaces Author George Kamberov
ISBN-10 9780821819289
Release 2002
Pages 138
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Many problems in pure and applied mathematics boil down to determining the shape of a surface in space or constructing surfaces with prescribed geometric properties. These problems range from classical problems in geometry, elasticity, and capillarity to problems in computer vision, medical imaging, and graphics. There has been a sustained effort to understand these questions, but many problems remain open or only partially solved. This book describes how to use quaternions and spinors to study conformal immersions of Riemann surfaces into $\Bbb R^3$. The first part develops the necessary quaternionic calculus on surfaces, its application to surface theory and the study of conformal immersions and spinor transforms. The integrability conditions for spinor transforms lead naturally to Dirac spinors and their application to conformal immersions. The second part presents a complete spinor calculus on a Riemann surface, the definition of a conformal Dirac operator, and a generalized Weierstrass representation valid for all surfaces. This theory is used to investigate first, to what extent a surface is determined by its tangent plane distribution, and second, to what extent curvature determines the shape. The book is geared toward graduate students and researchers interested in differential geometry and geometric analysis and their applications in computer vision and computer graphics.



Geometric Algebra for Computer Graphics

Geometric Algebra for Computer Graphics Author John Vince
ISBN-10 9781846289972
Release 2008-02-10
Pages 256
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Geometric algebra (a Clifford Algebra) has been applied to different branches of physics for a long time but is now being adopted by the computer graphics community and is providing exciting new ways of solving 3D geometric problems. The author tackles this complex subject with inimitable style, and provides an accessible and very readable introduction. The book is filled with lots of clear examples and is very well illustrated. Introductory chapters look at algebraic axioms, vector algebra and geometric conventions and the book closes with a chapter on how the algebra is applied to computer graphics.



New Trends in Computer Graphics

New Trends in Computer Graphics Author Nadia Magnenat-Thalmann
ISBN-10 9783642834929
Release 2012-12-06
Pages 682
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New Trends in Computer Graphics contains a selection of research papers submitted to Computer Graphics International '88 (COl '88). COl '88 is the Official Annual Conference of the Computer Graphics Society. Since 1982, this conference ha~ been held in Tokyo. This year, it is taking place in Geneva, Switzerland. In 1989, it will be held in Leeds, U. K. , in 1990 in Singapore, in 1991 in U. S. A. and in 1992 in Montreal, Canada. Over 100 papers were submitted to CGI '88 and 61 papers were selected by the International Program Committee. Papers have been grouped into 6 chapters. The flrst chapter is dedicated to Computer Animation because it deals with all topics presented in the other chapters. Several animation systems are described as well as speciflc subjects like 3D character animation, quaternions and splines. The second chapter is dedicated to papers on Image Synthesis, il1 particular new shading models and new algorithms for ray tracing are presented. Chapter 3 presents several algorithms for geometric modeling and new techniques for the creation and manipulation of curves, surfaces and solids and their applications to CAD. In Chapter 4, an important topic is presented: the specification of graphics systems and images using l~nguages and user-interfaces. The last two chapters are devoted to applications in sciences, medicine, engineering, art and business.



3D Math Primer for Graphics and Game Development 2nd Edition

3D Math Primer for Graphics and Game Development  2nd Edition Author Fletcher Dunn
ISBN-10 9781568817231
Release 2011-11-02
Pages 846
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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.



Calculus for Computer Graphics

Calculus for Computer Graphics Author John Vince
ISBN-10 9781447154662
Release 2013-08-27
Pages 227
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Students studying computer animation and computer games have to be familiar with geometry, matrices, vectors, rotation transforms, quaternions, curves and surfaces, and as computer graphics software becomes increasingly sophisticated, calculus is also being used to resolve its associated problems. The author draws upon his experience in teaching mathematics to undergraduates to make calculus appear no more challenging than any other branch of mathematics. He introduces the subject by examining how functions depend upon their independent variables, and then derives the appropriate mathematical underpinning and definitions. This gives rise to a function’s derivative and its antiderivative, or integral. Using the idea of limits, the reader is introduced to derivatives and integrals of many common functions. Other chapters address higher-order derivatives, partial derivatives, Jacobians, vector-based functions, single, double and triple integrals, with numerous worked examples, and over a hundred illustrations. Calculus for Computer Graphics complements the author’s other books on mathematics for computer graphics, and assumes that the reader is familiar with everyday algebra, trigonometry, vectors and determinants. After studying this book, the reader should understand calculus and its application within the world of computer games and animation.



Mathematics for 3D Game Programming and Computer Graphics Third Edition

Mathematics for 3D Game Programming and Computer Graphics  Third Edition Author Eric Lengyel
ISBN-10 9781435458871
Release 2012
Pages 576
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This updated third edition addresses the mathematical skills that a programmer needs to develop a 3D game engine and computer graphics for professional-level games. MATHEMATICS FOR 3D GAME PROGRAMMING & COMPUTER GRAPHICS, THIRD EDITION is suitable for adv



Essential Mathematics for Computer Graphics fast

Essential Mathematics for Computer Graphics fast Author John Vince
ISBN-10 1852333804
Release 2001-01-01
Pages 229
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This book provides a quick, concise introduction to the most important mathematics used in computer graphics. Vince makes the concepts easy to understand, so that non-mathematicians can grasp the math that lies behind computer animation.