[...]... refresher to readers whose educations in the topics are many years removed from them Interactive Demo Applications 5 Interactive Demo Applications Demo Name Three-dimensional games and graphics are, by their nature, not only visual but dynamic While figures are indeed a welcome necessity in a book about 3D applications, interactive demos can be even more important It is difficult to truly understand such... programmers often need to learn or refresh the basic mathematics behind 3D games before continuing on to learn the applications of these principles to rendering and animation On the other hand, university students entering (or hoping to enter) the 3D games industry often ask what material they need to know in order to be prepared to work on these games Younger students often ask what courses they should... background for a programmer in the industry Recent graduates, on the other hand, often ask how their computer graphics knowledge best relates to the way games are developed for today’s computers and game consoles We have designed this book to provide something for each of these groups of readers We attempt to provide readers with a conceptual understanding of the mathematics needed to create 3D games, ... performance in 3D applications Part II, Rendering Chapters 5 through 8 explain the so-called rendering pipeline, from the way we represent visible objects in 3D games to the methods used to draw these objects to the display A mixture of concepts, mathematics, and implementations, these chapters begin to show the direct applicability of the mathematics introduced in the first four chapters to the 3D games. .. 3 Affine Transformations 3.1 3.2 3.3 3.4 Introduction 107 Affine Transformations 108 3.2.1 Definition 108 3.2.2 Representation 109 Standard Affine Transformations 113 3.3.1 Translation 113 3.3.2 Rotation 115 3.3.3 Scaling 124 3.3.4 Reflection 126 3.3.5 Shear 130 3.3.6 Applying an Affine Transformation Around an Arbitrary Point 132 3.3.7 Transforming Plane Normals 134 Using Affine Transformations 135...Contents Preface Introduction xxi 1 The (Continued) Rise of 3D Games 1 How to Read this Book 2 Part I, Core Mathematics 2 Part II, Rendering 3 Part III, Animation 4 Part IV, Simulation 4 Appendices 4 Interactive Demo Applications 5 Support Libraries 5 Math Libraries 6 Engine and Rendering Libraries 6 References and Further Reading 7 Part I Core Mathematics Chapter 1 Vectors and Points 1.1 1.2 11 Introduction... looking for someone to write just that book for Morgan Kaufmann At the same time, Lars was presenting some of the basics of rendering on handheld devices as part of a SIGGRAPH course Jim and Lars discussed the fact that handheld 3D rendering had brought back some of the “lost arts” of 3D programming, and that this might be included in a book on mathematics for game programming Thus, a co-authorship was formed... with IvMath) in other applications Finally, the simulation demos use a shared library called IvCollision, which implements basic object intersection (collision) data structures and algorithms Building on the IvMath library, this set of classes and functions forms not only the basis for the later demos in the book but also is an excellent starting point for experimentation with other forms of object collision... Precision 193 4.8.2 Performance 194 4.8.3 IEEE Specification Compliance 197 Code 198 Chapter Summary 199 Part II RENDERING Chapter 5 Viewing and Projection 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Introduction 203 The View Frame and View Transformation 204 5.2.1 Defining a Virtual Camera 204 5.2.2 Controlling the Camera 207 5.2.3 Constructing the View Transformation 210 Projective Transformation 211 5.3.1 Definition... Summary 63 Chapter 2 Linear Transformations and Matrices 2.1 2.2 2.3 Introduction 65 Linear Transformations 66 2.2.1 Definitions 66 2.2.2 Null Space and Range 67 2.2.3 Linear Transformations and Basis Vectors 69 Matrices 71 2.3.1 Introduction to Matrices 71 2.3.2 Simple Operations 73 2.3.3 Vector Representation 75 Block Matrices 75 2.3.4 2.3.5 Matrix Product 77 2.3.6 Transforming Vectors 79 65 Contents . goal is for the developer to be able to implement real systems from the fundamental ideas, whether it be for games or for other applications. Essential Mathematics for Games and Interactive Applications: A. University, co-author of Level of Detail for 3D Graphics Essential Mathematics for Games and Interactive Applications A Programmer’s Guide The Morgan Kaufmann Series in Interactive 3D Technology Series. excellent framework for developing robust and fast applications. —Ian Ashdown, President, ByHeart Consultants Limited With Essential Mathematics for Games and Interactive Applications, Van Verth