Mathematics for 3D Game Programming & Computer Graphics

Summary

This invaluable resource illustrates the mathematics that a game programmer would need to develop a professional-quality 3D engine. This book starts at a fairly basic level in each of several areas such as vector geometry, modern algebra, and physics, and then progresses to somewhat more advanced topics. Particular attention is given to derivations of key results, ensuring that the reader is not forced to endure “gaps” in the theory. The book will discuss applications in the context of the OpenGL architecture. The book assumes basic understanding of matrix algebra, trigonometry, and calculus. 

Table of Contents

PART I — MATHEMATICAL FOUNDATIONS. SECTION 1 — Vector and Matrix Geometry. Properties of vector spaces. Dot products and cross products. Homogeneous coordinates. Representing planes as 4D vectors. Coordinate transforma-tions. Perspective projections. SECTION 2 — Algebra and Calculus. Root finding, Newton’s method. Analytic solution to quadratic, cubic, and quartic equations. Ray intersections with surfaces. Calculating normals and tangents. Contravariant and covariant vectors. Differential equations. SECTION 3 — Modern Algebra.Introduction to group theory. Rotation groups. Homomorphisms. Complex numbers. Quaternions. Spherical linear interpolation. PART II — RENDERING. SECTION 1 — Ray Tracing. Illumination equations. Reflection and refraction. Bump mapping. SECTION 2 — Rasterization. Conventional rasterization pipeline. Modern per-pixel lighting methods. Environment mapping. Shadow rendering techniques.SECTION 3 — Polygonal Techniques. Polygon clipping. Portal clipping. T-junction elimination. Decaling and depth offset. SECTION 4 — Visibility Determination. Bounding volumes. View frustum/portal intersect tests. Plane visibility. BSP trees. PART III — PHYSICS. SECTION 1 — Linear motion. Projectile motion. Position functions. Collision detection. Frictional forces. Elastic collisions SECTION 2 — Rotational motion. Moment of inertia. Angular momentum and torque. Centrifugal force. Coriolis force.