[...]... to practical applications in 3D game programming and computer graphics After analyzing lines and planes in 3D space, we introduce the view frustum and its relationship to the virtual camera This chapter includes topics such as field of view, perspectivecorrect interpolation, and projection matrices Chapter 6: Ray Tracing Ray tracing methods are useful in many areas of game programming, including light... portion of the book with a thorough review of vector quantities and their properties Vectors are of fundamental importance in the study of 3D computer graphics, and we make extensive use of operations such as the dot product and cross product throughout the book Chapter 3: Matrices An understanding of matrices is another basic necessity of 3D game programming This chapter discusses elementary concepts such... surfaces and includes a related method for performing vertex depth offset Other topics include billboarding techniques used for various special effects, a polygon reduction technique, T-junction elimination, and polygon triangulation Chapter 10: Shadows This chapter discusses shadow-casting sets and the two prominent methods for generating shadows in a real-time application, shadow mapping and stencil... derives the Taylor series and reviews power series representations for many common functions Appendix E: Answers to Exercises This appendix contains the answer to every exercise in the book having a solution that can be represented by a mathematical expression Website and Code Listings The official website for this book can be found at the following address: http://www.mathfor3dgameprogramming.com/ All... downloaded from this website Some of the listings make use of simple structures such as Triangle and Edge or mathematical classes such as Vector3D and Matrix3D The source code for these can also be found on the website The language used for all code that is intended to run on the CPU is standard C++ Vertex shaders and fragment shaders that are intended to run on the GPU use the OpenGL Shading Language (GLSL)... line-of-sight determination, and collision detection This chapter begins with analytical and numerical rootfinding techniques, and then presents methods for intersecting rays with common geometrical objects Finally, calculation of reflection and refraction vectors is discussed Chapter 7: Lighting and Shading Chapter 7 discusses a wide range of topics related to illumination and shading methods We begin... eigenvectors and diagonalization, which are required later in the book For completeness, this chapter goes into a lot of detail to prove various matrix properties, but an understanding of those details is not essential when using matrices later in the book, so the uninterested reader may skip over the proofs Chapter 4: Transforms In Chapter 4, we investigate matrices as a tool for performing transformations... the GPU can understand A 3D graphics driver usually implements OpenGL functions directly to minimize the overhead of issuing rendering commands The block diagram shown in Figure 1.2 illustrates the communications that take place between the CPU and GPU A 3D graphics board has its own memory core, which is commonly called VRAM (Video Random Access Memory) The GPU may store any information in VRAM, but... is possible to transform vertices from object space directly into camera space by concatenating the matrices representing the transformations from object space to world space and from world space to camera space The product of these transformations is called the model-view transformation Once a model’s vertices have been transformed into camera space, they undergo a projection transformation that has... complex numbers and discusses the properties that are used elsewhere in the book Website and Code Listings  Appendix B: Trigonometry Reference Appendix B reviews the trigonometric functions and quickly derives many formulas and identities that are used throughout this book Appendix C: Coordinate Systems Appendix C provides a brief overview of Cartesian coordinates, cylindrical coordinates, and spherical . Spain • United Kingdom • United States Mathematics for 3D Game Programming and Computer Graphics, Third Edition By Eric Lengyel Publisher and General Manager, Course Technology PTR:.