Visualizing Quaternions

Author: Andrew J. Hanson
Editor: Elsevier
ISBN: 0080474772
Size: 20,80 MB
Format: PDF, ePub, Docs
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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.

Visualizing Quaternions

Author: Andrew Hanson ((Andrew J.))
Editor:
ISBN:
Size: 17,40 MB
Format: PDF, ePub, Docs
Read: 796
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Modern Differential Geometry Of Curves And Surfaces With Mathematica Third Edition

Author: Elsa Abbena
Editor: CRC Press
ISBN: 9781584884484
Size: 13,60 MB
Format: PDF
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Presenting theory while using Mathematica in a complementary way, Modern Differential Geometry of Curves and Surfaces with Mathematica, the third edition of Alfred Gray’s famous textbook, covers how to define and compute standard geometric functions using Mathematica for constructing new curves and surfaces from existing ones. Since Gray’s death, authors Abbena and Salamon have stepped in to bring the book up to date. While maintaining Gray's intuitive approach, they reorganized the material to provide a clearer division between the text and the Mathematica code and added a Mathematica notebook as an appendix to each chapter. They also address important new topics, such as quaternions. The approach of this book is at times more computational than is usual for a book on the subject. For example, Brioshi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing curvature. Another part of Mathematica that can be used effectively in differential geometry is its special function library, where nonstandard spaces of constant curvature can be defined in terms of elliptic functions and then plotted. Using the techniques described in this book, readers will understand concepts geometrically, plotting curves and surfaces on a monitor and then printing them. Containing more than 300 illustrations, the book demonstrates how to use Mathematica to plot many interesting curves and surfaces. Including as many topics of the classical differential geometry and surfaces as possible, it highlights important theorems with many examples. It includes 300 miniprograms for computing and plotting various geometric objects, alleviating the drudgery of computing things such as the curvature and torsion of a curve in space.

Rethinking Quaternions

Author: Ron Goldman
Editor: Morgan & Claypool Publishers
ISBN: 1608454207
Size: 16,95 MB
Format: PDF, Kindle
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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
Editor: CRC Press
ISBN: 1439869812
Size: 19,37 MB
Format: PDF, Mobi
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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.

Holistic Game Development With Unity

Author: Penny de Byl
Editor: Taylor & Francis
ISBN: 1136147179
Size: 13,71 MB
Format: PDF, ePub
Read: 711
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The independent developer has ascended, and the new business model demands agility. You have to be able to work on all aspects of game creation, and your team's game will publish directly to platforms like Android, iPhone, and Facebook. You'll use Unity, the hottest game engine out there, to do it. In order to earn your place on the elite development team, you must master both sides of the development coin: art and programming. Holistic Game Development with Unity is an authoritative guide to creating games in Unity. Taking you through game design, programming, and art, Penny de Byl uses a holistic approach to equip you with the multidisciplinary skills you need for the independent games industry. With this book, you will master essential digital art and design principles while learning the programming skills necessary to build interactivity into your games. The tutorials will put these skills into action. The companion website offers: source code for completed projects from the book, art assets, instructional videos, a forum, author blog and lesson plans and challenge questions for professors. Examines art and programming in unison-the only one-stop shop for individual developers and small teams looking to tackle both tasks.