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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

Until recently, 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.

• Explains GA as a natural extension of linear algebra and conveys its significance for 3D programming of geometry in graphics, vision, and robotics.
• Systematically explores the concepts and techniques that are key to representing elementary objects and geometric operators using GA.
• Covers in detail the conformal model, a convenient way to implement 3D geometry using a 5D representation space.
• Presents effective approaches to making GA an integral part of your programming.
• Includes numerous drills and programming exercises helpful for both students and practitioners.
• Companion web site includes links to GAViewer, a program that will allow you to interact with many of the 3D figures in the book, and Gaigen 2, the platform for the instructive programming exercises that conclude each chapter.

Geometric Measure Theory: A Beginner's Guide, Fifth Edition provides the framework readers need to understand the structure of a crystal, a soap bubble cluster, or a universe.

The book is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Brevity, clarity, and scope make this classic book an excellent introduction to more complex ideas from geometric measure theory and the calculus of variations for beginning graduate students and researchers.

Morgan emphasizes geometry over proofs and technicalities, providing a fast and efficient insight into many aspects of the subject, with new coverage to this edition including topical coverage of the Log Convex Density Conjecture, a major new theorem at the center of an area of mathematics that has exploded since its appearance in Perelman's proof of the Poincaré conjecture, and new topical coverage of manifolds taking into account all recent research advances in theory and applications.

• Focuses on core geometry rather than proofs, paving the way to fast and efficient insight into an extremely complex topic in geometric structures
• Enables further study of more advanced topics and texts
• Demonstrates in the simplest possible way how to relate concepts of geometric analysis by way of algebraic or topological techniques
• Contains full topical coverage of The Log-Convex Density Conjecture
• Comprehensively updated throughout

Geometric Morphometrics for Biologists is an introductory textbook for a course on geometric morphometrics, written for graduate students and upper division undergraduates, covering both theory of shape analysis and methods of multivariate analysis. It is designed for students with minimal math background; taking them from the process of data collection through basic and more advanced statistical analyses. Many examples are given, beginning with simple although realistic case-studies, through examples of complex analyses requiring several different kinds of methods. The book also includes URL’s for free software and step-by-step instructions for using the software.

* Accessible, student-tested introduction to sophisticated methods of biological shape analysis
* Detailed instructions for conducting analyses with freely available, easy to use software
* Numerous illustrations; including graphical presentations of important theoretical concepts and demonstrations of alternative approaches to presenting results
* Many realistic examples, both simple and complex, from on-going research
* Comprehensive glossary of technical terms

The first edition of Geometric Morphometrics for Biologists has been the primary resource for teaching modern geometric methods of shape analysis to biologists who have a stronger background in biology than in multivariate statistics and matrix algebra. These geometric methods are appealing to biologists who approach the study of shape from a variety of perspectives, from clinical to evolutionary, because they incorporate the geometry of organisms throughout the data analysis. The second edition of this book retains the emphasis on accessible explanations, and the copious illustrations and examples of the first, updating the treatment of both theory and practice. The second edition represents the current state-of-the-art and adds new examples and summarizes recent literature, as well as provides an overview of new software and step-by-step guidance through details of carrying out the analyses.

• Contains updated coverage of methods, especially for sampling complex curves and 3D forms and a new chapter on applications of geometric morphometrics to forensics
• Offers a reorganization of chapters to streamline learning basic concepts
• Presents detailed instructions for conducting analyses with freely available, easy to use software
• Provides numerous illustrations, including graphical presentations of important theoretical concepts and demonstrations of alternative approaches to presenting results

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.

Engineering drawing is the "instrument of communication" upon which the designer must place all information necessary to define a new product. Computer-aided design (CAD) courses often involve teaching solid modelling software, and we view CAD as an engineering communication tool for manufacturing. As the technology of engineering design is in transition from paper drawings to solid models, its education must address the challenge of covering both technologies. Geometry of design integrates drafting technology based on experience with engineering design education. This workbook has evolved from the course “Computer-Aided Graphics and Design” at the University of Florida, and many pages of this textbook can be used for student assignments. In order to help students to familiarize themselves with the manufacturing field experience, most assignments are to be submitted in the form of complete working drawings of the parts and assembly. The first three chapters introduce basic engineering drawing definitions and practices. The following four chapters cover design and descriptive geometry, and subsequent chapters move on to dimensions, assembly line design and surface development.

Geometry with Trigonometry Second Edition is a second course in plane Euclidean geometry, second in the sense that many of its basic concepts will have been dealt with at school, less precisely. It gets underway with a large section of pure geometry in Chapters 2 to 5 inclusive, in which many familiar results are efficiently proved, although the logical frame work is not traditional. In Chapter 6 there is a convenient introduction of coordinate geometry in which the only use of angles is to handle the perpendicularity or parallelism of lines. Cartesian equations and parametric equations of a line are developed and there are several applications. In Chapter 7 basic properties of circles are developed, the mid-line of an angle-support, and sensed distances. In the short Chaper 8 there is a treatment of translations, axial symmetries and more generally isometries. In Chapter 9 trigonometry is dealt with in an original way which e.g. allows concepts such as clockwise and anticlockwise to be handled in a way which is not purely visual. By the stage of Chapter 9 we have a context in which calculus can be developed. In Chapter 10 the use of complex numbers as coordinates is introduced and the great conveniences this notation allows are systematically exploited. Many and varied topics are dealt with , including sensed angles, sensed area of a triangle, angles between lines as opposed to angles between co-initial half-lines (duo-angles). In Chapter 11 various convenient methods of proving geometrical results are established, position vectors, areal coordinates, an original concept mobile coordinates. In Chapter 12 trigonometric functions in the context of calculus are treated.

New to this edition:

• The second edition has been comprehensively revised over three years
• Errors have been corrected and some proofs marginally improved
• The substantial difference is that Chapter 11 has been significantly extended, particularly the role of mobile coordinates, and a more thorough account of the material is given

• Provides a modern and coherent exposition of geometry with trigonometry for many audiences across mathematics
• Provides many geometric diagrams for a clear understanding of the text and includes problem exercises for many chapters
• Generalizations of this material, such as to solid euclidean geometry and conic sections, when combined with calculus, would lead to applications in science, engineering, and elsewhere