Foundations of Game Engine Development, Volume 1: Mathematics

by Eric Lengyel

The first volume of Foundations of Game Engine Development discusses the mathematics needed by engineers who work on games or other types of virtual simulations. The book begins with conventional treatments of topics such as linear algebra, transforms, and geometry. Then, it introduces Grassmann algebra and geometric algebra to provide a much deeper understanding of the subject matter and highlight the places where traditional arithmetic with vectors, matrices, quaternions, etc., isn't quite correct. Includes exercises.Chapter 1: Vectors and Matrices1.1 Vector Fundamentals1.2 Basic Vector Operations1.2.1 Magnitude and Scalar Multiplication1.2.2 Addition and Subtraction1.3 Matrix Fundamentals1.4 Basic Matrix Operations1.4.1 Addition, Subtraction, and Scalar Multiplication1.4.2 Matrix Multiplication1.5 Vector Multiplication1.5.1 Dot Product1.5.2 Cross Product1.5.3 Scalar Triple Product1.6 Vector Projection1.7 Matrix Inversion1.7.1 Identity Matrices1.7.2 Determinants1.7.3 Elementary Matrices1.7.4 Inverse Calculation1.7.5 Inverses of Small MatricesChapter 2: Transforms2.1 Coordinate Spaces2.1.1 Transformation Matrices2.1.2 Orthogonal Transforms2.1.3 Transform Composition2.2 Rotations2.2.1 Rotation About a Coordinate Axis2.2.2 Rotation About an Arbitrary Axis2.3 Reflections2.4 Scales2.5 Skews2.6 Homogeneous Coordinates2.7 Quaternions2.7.1 Quaternion Fundamentals2.7.2 Rotations With QuaternionsChapter 3: Geometry3.1 Triangle Meshes3.2 Normal Vectors3.2.1 Calculating Normal Vectors3.2.2 Transforming Normal Vectors3.3 Lines and Rays3.3.1 Parametric Lines3.3.2 Distance Between a Point and a Line3.3.3 Distance Between Two Lines3.4 Planes3.4.1 Implicit Planes3.4.2 Distance Between a Point and a Plane3.4.3 Reflection Through a Plane3.4.4 Intersection of a Line and a Plane3.4.5 Intersection of Three Planes3.4.6 Intersection of Two Planes3.4.7 Transforming Planes3.5 Plücker Coordinates3.5.1 Implicit Lines3.5.2 Homogeneous Formulas3.5.3 Transforming LinesChapter 4: Advanced Algebra4.1 Grassmann Algebra4.1.1 Wedge Product4.1.2 Bivectors4.1.3 Trivectors4.1.4 Algebraic Structure4.1.5 Complements4.1.6 Antivectors4.1.7 Antiwedge Product4.2 Projective Geometry4.2.1 Lines4.2.2 Planes4.2.3 Join and Meet4.2.4 Line Crossing4.2.5 Plane Distance4.2.6 Summary and Implementation4.3 Matrix Inverses4.4 Geometric Algebra4.4.1 Geometric Product4.4.2 Vector Division4.4.3 Rotors4.5 Conclusion

Genres: Programming, Mathematics, Games, Technology, Computer Science, Computers, Textbooks

188 pages, Paperback

Published September 11, 2016

Reviews

[Jose Lopez · Rating 5 out of 5]

A great book for anyone interested in or already part of the 3D programming world.

It is a relatively short book, but it still manages to explain everything in a clear and concise manner.

Last chapter is particularly interesting. It offers a great look into Grassmann algebra, a subject that is not easy to find anywhere else.

My only comment, if any, is that it would have been nice if some brief examples or practical explanations had been provided, of how Grassmann algebra could be applied to 3D graphics. I will definitely have to go back at some point and re-read Chapter 4 to fully grasp it.

Definitely a must have!

[Lohit Br · Rating 5 out of 5]

Bill Ericson's Foundations of Game Engine Development, Volume 1: Mathematics is an essential resource for aspiring and experienced game developers alike. This comprehensive book covers key mathematical concepts such as linear algebra, geometry, and calculus, all within the context of game engine development. Ericson's clear writing and practical examples make complex topics accessible, while exercises reinforce learning.

Whether you're new to game development or looking to deepen your knowledge, this book provides a solid foundation for understanding the mathematics that drive game engines. A must-read for anyone serious about a career in the gaming industry.

For those interested in game development services, check out BR Softech for expert solutions to bring your game ideas to life!

[Qwiff · Rating 5 out of 5]

What makes this book stand out from other game math literature, is that it weaves an engaging narrative around its subject matter. Instead of treating the topics separately, like other books, it connects them right from the start. For example, chapter 1 switches between vectors and matrices, gradually introducing their applications along the way.

The chapter on Grassman algebra was an eye-opener for me.