# Geometric Algebra for Computer Science

An Object-Oriented Approach to Geometry

Author: Leo Dorst,Daniel Fontijne,Stephen Mann

Publisher: Elsevier

ISBN: 0080553109

Category: Computers

Page: 664

View: 8962

Until recently, almost all of the interactions between objects in virtual 3D worlds have been based on calculations performed using linear algebra. Linear algebra relies heavily on coordinates, however, which can make many geometric programming tasks very specific and complex-often a lot of effort is required to bring about even modest performance enhancements. Although linear algebra is an efficient way to specify low-level computations, it is not a suitable high-level language for geometric programming. Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhancing way to represent the geometry of 3D objects in computer programs. In this book you will find an introduction to GA that will give you a strong grasp of its relationship to linear algebra and its significance for your work. You will learn how to use GA to represent objects and perform geometric operations on them. And you will begin mastering proven techniques for making GA an integral part of your applications in a way that simplifies your code without slowing it down. * The first book on Geometric Algebra for programmers in computer graphics and entertainment computing * Written by leaders in the field providing essential information on this new technique for 3D graphics * This full colour book includes a website with GAViewer, a program to experiment with GA

# Geometric Algebra: An Algebraic System for Computer Games and Animation

Author: John A. Vince

Publisher: Springer Science & Business Media

ISBN: 9781848823792

Category: Computers

Page: 195

View: 2432

Geometric algebra is still treated as an obscure branch of algebra and most books have been written by competent mathematicians in a very abstract style. This restricts the readership of such books especially by programmers working in computer graphics, who simply want guidance on algorithm design. Geometric algebra provides a unified algebraic system for solving a wide variety of geometric problems. John Vince reveals the beauty of this algebraic framework and communicates to the reader new and unusual mathematical concepts using colour illustrations, tabulations, and easy-to-follow algebraic proofs. The book includes many worked examples to show how the algebra works in practice and is essential reading for anyone involved in designing 3D geometric algorithms.

# A New Approach to Differential Geometry using Clifford's Geometric Algebra

Author: John Snygg

Publisher: Springer Science & Business Media

ISBN: 081768283X

Category: Mathematics

Page: 465

View: 6889

Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.

# Geometric Tools for Computer Graphics

Author: Philip Schneider,David H. Eberly

Publisher: Elsevier

ISBN: 9780080478029

Category: Computers

Page: 1056

View: 3387

Do you spend too much time creating the building blocks of your graphics applications or finding and correcting errors? Geometric Tools for Computer Graphics is an extensive, conveniently organized collection of proven solutions to fundamental problems that you'd rather not solve over and over again, including building primitives, distance calculation, approximation, containment, decomposition, intersection determination, separation, and more. If you have a mathematics degree, this book will save you time and trouble. If you don't, it will help you achieve things you may feel are out of your reach. Inside, each problem is clearly stated and diagrammed, and the fully detailed solutions are presented in easy-to-understand pseudocode. You also get the mathematics and geometry background needed to make optimal use of the solutions, as well as an abundance of reference material contained in a series of appendices. Features Filled with robust, thoroughly tested solutions that will save you time and help you avoid costly errors. Covers problems relevant for both 2D and 3D graphics programming. Presents each problem and solution in stand-alone form allowing you the option of reading only those entries that matter to you. Provides the math and geometry background you need to understand the solutions and put them to work. Clearly diagrams each problem and presents solutions in easy-to-understand pseudocode. Resources associated with the book are available at the companion Web site www.mkp.com/gtcg. * Filled with robust, thoroughly tested solutions that will save you time and help you avoid costly errors. * Covers problems relevant for both 2D and 3D graphics programming. * Presents each problem and solution in stand-alone form allowing you the option of reading only those entries that matter to you. * Provides the math and geometry background you need to understand the solutions and put them to work. * Clearly diagrams each problem and presents solutions in easy-to-understand pseudocode. * Resources associated with the book are available at the companion Web site www.mkp.com/gtcg.

# Foundations of Geometric Algebra Computing

Author: Dietmar Hildenbrand

Publisher: Springer Science & Business Media

ISBN: 3642317944

Category: Computers

Page: 196

View: 4820

The author defines “Geometric Algebra Computing” as the geometrically intuitive development of algorithms using geometric algebra with a focus on their efficient implementation, and the goal of this book is to lay the foundations for the widespread use of geometric algebra as a powerful, intuitive mathematical language for engineering applications in academia and industry. The related technology is driven by the invention of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and by the recent progress in parallel processing, and with the specific conformal geometric algebra there is a growing community in recent years applying geometric algebra to applications in computer vision, computer graphics, and robotics. This book is organized into three parts: in Part I the author focuses on the mathematical foundations; in Part II he explains the interactive handling of geometric algebra; and in Part III he deals with computing technology for high-performance implementations based on geometric algebra as a domain-specific language in standard programming languages such as C++ and OpenCL. The book is written in a tutorial style and readers should gain experience with the associated freely available software packages and applications. The book is suitable for students, engineers, and researchers in computer science, computational engineering, and mathematics.

# Geometric Algebra for Computer Graphics

Author: John Vince

Publisher: Springer Science & Business Media

ISBN: 1846289971

Category: Computers

Page: 256

View: 9021

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.

# Geometric Algebra with Applications in Engineering

Author: Christian Perwass

Publisher: Springer Science & Business Media

ISBN: 3540890688

Category: Computers

Page: 386

View: 8526

The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the mathematical structure of geometric algebra together with its descriptive power will result in intuitive and more robust algorithms. This book examines all aspects essential for a successful application of geometric algebra: the theoretical foundations, the representation of geometric constraints, and the numerical estimation from uncertain data. Formally, the book consists of two parts: theoretical foundations and applications. The first part includes chapters on random variables in geometric algebra, linear estimation methods that incorporate the uncertainty of algebraic elements, and the representation of geometry in Euclidean, projective, conformal and conic space. The second part is dedicated to applications of geometric algebra, which include uncertain geometry and transformations, a generalized camera model, and pose estimation. Graduate students, scientists, researchers and practitioners will benefit from this book. The examples given in the text are mostly recent research results, so practitioners can see how to apply geometric algebra to real tasks, while researchers note starting points for future investigations. Students will profit from the detailed introduction to geometric algebra, while the text is supported by the author's visualization software, CLUCalc, freely available online, and a website that includes downloadable exercises, slides and tutorials.

# Jim Blinn's Corner: Notation, Notation, Notation

Author: Jim Blinn

Publisher: Elsevier

ISBN: 0080509606

Category: Computers

Page: 336

View: 1402

The third entry in the Jim Blinn's Corner series, this is, like the others, a handy compilation of selected installments of his influential column. But here, for the first time, you get the "Director's Cut" of the articles: revised, expanded, and enhanced versions of the originals. What's changed? Improved mathematical notation, more diagrams, new solutions. What remains the same? All the things you've come to rely on: straight answers, irreverent style, and innovative thinking. This is Jim Blinn at his best - now even better. Features 21 expanded and updated installments of "Jim Blinn's Corner," dating from 1995 to 2001, and never before published in book form Includes "deleted scenes"—tangential explorations that didn't make it into the original columns Details how Blinn represented planets in his famous JPL flyby animations Explores a wide variety of other topics, from the concrete to the theoretical: assembly language optimization for parallel processors, exotic usage of C++ template instantiation, algebraic geometry, a graphical notation for tensor contraction, and his hopes for a future world

# Clifford (Geometric) Algebras

with applications to physics, mathematics, and engineering

Author: William Baylis

Publisher: Springer Science & Business Media

ISBN: 1461241049

Category: Science

Page: 517

View: 4590

This volume is an outgrowth of the 1995 Summer School on Theoretical Physics of the Canadian Association of Physicists (CAP), held in Banff, Alberta, in the Canadian Rockies, from July 30 to August 12,1995. The chapters, based on lectures given at the School, are designed to be tutorial in nature, and many include exercises to assist the learning process. Most lecturers gave three or four fifty-minute lectures aimed at relative novices in the field. More emphasis is therefore placed on pedagogy and establishing comprehension than on erudition and superior scholarship. Of course, new and exciting results are presented in applications of Clifford algebras, but in a coherent and user-friendly way to the nonspecialist. The subject area of the volume is Clifford algebra and its applications. Through the geometric language of the Clifford-algebra approach, many concepts in physics are clarified, united, and extended in new and sometimes surprising directions. In particular, the approach eliminates the formal gaps that traditionally separate clas sical, quantum, and relativistic physics. It thereby makes the study of physics more efficient and the research more penetrating, and it suggests resolutions to a major physics problem of the twentieth century, namely how to unite quantum theory and gravity. The term "geometric algebra" was used by Clifford himself, and David Hestenes has suggested its use in order to emphasize its wide applicability, and b& cause the developments by Clifford were themselves based heavily on previous work by Grassmann, Hamilton, Rodrigues, Gauss, and others.

# 3D Game Engine Design

A Practical Approach to Real-Time Computer Graphics

Author: David H. Eberly

Publisher: CRC Press

ISBN: 1482267306

Category: Art

Page: 1040

View: 9077

A major revision of the international bestseller on game programming! Graphics hardware has evolved enormously in the last decade. Hardware can now be directly controlled through techniques such as shader programming, which requires an entirely new thought process of a programmer. 3D Game Engine Design, Second Edition shows step-by-step how to make a shader-based graphics engine and how to tame the new technology. Much new material has been added, including more than twice the coverage of the essential techniques of scene graph management, as well as new methods for managing memory usage in the new generation of game consoles and portable game players. There are expanded discussions of collision detection, collision avoidance, and physics - all challenging subjects for developers.

# Real-Time Collision Detection

Author: Christer Ericson

Publisher: CRC Press

ISBN: 1558607323

Category: Art

Page: 632

View: 8943

Written by an expert in the game industry, Christer Ericson's new book is a comprehensive guide to the components of efficient real-time collision detection systems. The book provides the tools and know-how needed to implement industrial-strength collision detection for the highly detailed dynamic environments of applications such as 3D games, virtual reality applications, and physical simulators. Of the many topics covered, a key focus is on spatial and object partitioning through a wide variety of grids, trees, and sorting methods. The author also presents a large collection of intersection and distance tests for both simple and complex geometric shapes. Sections on vector and matrix algebra provide the background for advanced topics such as Voronoi regions, Minkowski sums, and linear and quadratic programming. Of utmost importance to programmers but rarely discussed in this much detail in other books are the chapters covering numerical and geometric robustness, both essential topics for collision detection systems. Also unique are the chapters discussing how graphics hardware can assist in collision detection computations and on advanced optimization for modern computer architectures. All in all, this comprehensive book will become the industry standard for years to come.

# Principles of Digital Image Synthesis

Author: Andrew S. Glassner

Publisher: Elsevier

ISBN: 0080514758

Category: Computers

Page: 1600

View: 1500

Image synthesis, or rendering, is a field of transformation: it changes geometry and physics into meaningful images. Because the most popular algorithms frequently change, it is increasingly important for researchers and implementors to have a basic understanding of the principles of image synthesis. Focusing on theory, Andrew Glassner provides a comprehensive explanation of the three core fields of study that come together to form digital image synthesis: the human visual system, digital signal processing, and the interaction of matter and light. Assuming no more than a basic background in calculus, Glassner transforms his passion and expertise into a thorough presentation of each of these disciplines, and their elegant orchestration into modern rendering techniques such as radiosity and ray tracing.

# Curves and Surfaces for Computer Graphics

Author: David Salomon

Publisher: Springer Science & Business Media

ISBN: 0387284524

Category: Computers

Page: 460

View: 9230

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.

# Spatial Databases

With Application to GIS

Author: Philippe Rigaux,Michel O. Scholl,Agnès Voisard

Publisher: Morgan Kaufmann

ISBN: 9781558605886

Category: Computers

Page: 410

View: 3266

Spatial Databases is the first unified, in-depth treatment of special techniques for dealing with spatial data, particularly in the field of geographic information systems (GIS). This book surveys various techniques, such as spatial data models, algorithms, and indexing methods, developed to address specific features of spatial data that are not adequately handled by mainstream DBMS technology. The book also reviews commercial solutions to geographic data handling: ArcInfo, ArcView, and Smallworld GISs; and two extensions to the relational model, PostgreSQL and Oracle Spatial. The authors examine these underlying GIS technologies, assess their strengths and weaknesses, and consider specific uses for which each product is best suited. * Examines the strengths of various query languages and approaches to query processing. * Explains the use of computational geometry in spatial databases GISs, providing necessary background and an in-depth look at key algorithms. * Covers spatial access methods, including the R-tree and several space-driven structures, and is filled with dozens of helpful illustrations.

# Topological and Statistical Methods for Complex Data

Tackling Large-Scale, High-Dimensional, and Multivariate Data Spaces

Author: Janine Bennett,Fabien Vivodtzev,Valerio Pascucci

Publisher: Springer

ISBN: 3662449005

Category: Mathematics

Page: 297

View: 4652

This book contains papers presented at the Workshop on the Analysis of Large-scale, High-Dimensional, and Multi-Variate Data Using Topology and Statistics, held in Le Barp, France, June 2013. It features the work of some of the most prominent and recognized leaders in the field who examine challenges as well as detail solutions to the analysis of extreme scale data. The book presents new methods that leverage the mutual strengths of both topological and statistical techniques to support the management, analysis, and visualization of complex data. It covers both theory and application and provides readers with an overview of important key concepts and the latest research trends. Coverage in the book includes multi-variate and/or high-dimensional analysis techniques, feature-based statistical methods, combinatorial algorithms, scalable statistics algorithms, scalar and vector field topology, and multi-scale representations. In addition, the book details algorithms that are broadly applicable and can be used by application scientists to glean insight from a wide range of complex data sets.

# Visualizing Quaternions

Author: Andrew J. Hanson

Publisher: Elsevier

ISBN: 0080474772

Category: Mathematics

Page: 530

View: 9117

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.

# Geometric Algebra for Physicists

Author: Chris Doran,Anthony Lasenby

Publisher: Cambridge University Press

ISBN: 1139643142

Category: Science

Page: 578

View: 6718

Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory.

# Understanding Geometric Algebra

Hamilton, Grassmann, and Clifford for Computer Vision and Graphics

Author: Kenichi Kanatani

Publisher: CRC Press

ISBN: 1482259516

Category: Computers

Page: 208

View: 8026

Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics introduces geometric algebra with an emphasis on the background mathematics of Hamilton, Grassmann, and Clifford. It shows how to describe and compute geometry for 3D modeling applications in computer graphics and computer vision. Unlike similar texts, this book first gives separate descriptions of the various algebras and then explains how they are combined to define the field of geometric algebra. It starts with 3D Euclidean geometry along with discussions as to how the descriptions of geometry could be altered if using a non-orthogonal (oblique) coordinate system. The text focuses on Hamilton’s quaternion algebra, Grassmann’s outer product algebra, and Clifford algebra that underlies the mathematical structure of geometric algebra. It also presents points and lines in 3D as objects in 4D in the projective geometry framework; explores conformal geometry in 5D, which is the main ingredient of geometric algebra; and delves into the mathematical analysis of camera imaging geometry involving circles and spheres. With useful historical notes and exercises, this book gives readers insight into the mathematical theories behind complicated geometric computations. It helps readers understand the foundation of today’s geometric algebra.

# Geometric Algebra Computing

in Engineering and Computer Science

Author: Eduardo Bayro Corrochano,Gerik Scheuermann

Publisher: Springer Science & Business Media

ISBN: 1849961085

Category: Computers

Page: 526

View: 9308

This useful text offers new insights and solutions for the development of theorems, algorithms and advanced methods for real-time applications across a range of disciplines. Its accessible style is enhanced by examples, figures and experimental analysis.

# Computer Graphics

Theory and Practice

Author: Jonas Gomes,Luiz Velho,Mario Costa Sousa

Publisher: CRC Press

ISBN: 1439865574

Category: Computers

Page: 544

View: 2189

Computer Graphics: Theory and Practice provides a complete and integrated introduction to this area. The book only requires basic knowledge of calculus and linear algebra, making it an accessible introductory text for students. It focuses on conceptual aspects of computer graphics, covering fundamental mathematical theories and models and the inherent problems in implementing them. In so doing, the book introduces readers to the core challenges of the field and provides suggestions for further reading and studying on various topics. For each conceptual problem described, solution strategies are compared and presented in algorithmic form. This book, along with its companion Design and Implementation of 3D Graphics Systems, gives readers a full understanding of the principles and practices of implementing 3D graphics systems.