This completely updated second edition illustrates the mathematical concepts that a game programmer would need to develop a professional-quality 3D engine. Although the book is geared toward applications in game development, many of the topics appeal to general interests in 3D graphics. It starts at a fairly basic level in areas such as vector geometry and linear algebra, and then progresses to more advanced topics in 3D game programming such as illumination and visibility determination. Particular attention is given to derivations of key results, ensuring that the reader is not forced to endure gaps in the theory. The book assumes a working knowledge of trigonometry and calculus, but also includes sections that review the important tools used from these disciplines, such as trigonometric identities, differential equations, and Taylor series.

Little more than an outline

I was disappointed in how little explanations were included in this math book. It seemed more like a dictionary. It was as if someone listed all the math concepts that could be included and then put the list into a hard cover binder.
I did read some of the other reviews thinking I had missed something. One suggested spending a year on the book. But there is so little there. My college book on linear algebra did a much better job of explaining the material. Theorems were explained, samples worked out and 10 to 20 problems were given at the end of each section. This book as 50 problems for the entire book. This book spends eight pages on quaternions. Studying these pages over an extended period of time would not give the understanding you could get from a book like Kuipers’ Quaternions and Rotation Sequences. It took 75 pages but one is able to understand the system and how it relates to game programming.
With all the colleges now offering game programming degrees, their must be a book written by someone who is use to explaining the material. That person would know the best way to present the material and could anticipate questions and include the answers in his or her explanations.

An excellent book

I have personally used the C4 game engine (which is the commercial game engine developed by the author of this book), and can attest to its high quality. The engine is top notch, and so is this book.

This book is exactly what you’d expect - a description of the math involved in game (and especially graphics) development. It also includes a fair amount of additional content (not necessarily math-related, such as a description of the stencil shadowing algorithm).

The book is well-written, and does a good job of presenting fairly complex subject matter in a straight-forward manner. This is currently my favourite text book.

a disappointment

the second edition brings a new chapter on the graphics pipeline. Well done, except that it is extremely obscure, for those who already understood the arcanes of the graphics pipeline.
Chapter 5 on "ray tracing": 6 pages on root finding of quadratic, cubic and quartic polynomials;
to compare with Chapter 7: "Visibility determination": The spatial partitioning deserves solely 4 pages.
What’s the point on dwelling so much on something that is trivial and botching something that is hard to grasp and of paramount importance ?

One of a kind


As a professional 3D graphics programmer, I can not stress enough the quality of this book. This book covers 3D math fundamentals, algorithms, and it is complete with easy to understand (!) proofs. The math is difficult because there is so many problems to be solved in 3D (and they draw from many different branches of mathematics), but it is written in such a clear way that every topic is made approachable. Unlike esoteric Ph. D papers, you aren’t assumed to have any specific knowledge of math idioms or jargon. You simply need a decent grasp of college calculus and trigonometry to make the most of it. There are a few samples too to test your knowledge.

Expect to spend at least a solid year to really make the use of this book. Treat it as you would a two - three semester course in college. During this process, you’ll find yourself occasionally wanting to get more practice and referring to a respective book on it.

If you could only own two books for 3D programming, buy this first and buy Ericson’s book on collision detection next.

In summary, Eric Lengyel’s attention to detail and mastery of 3D math / algorithms really shines and this book is an example of it.

One of a kind



As a professional 3D graphics programmer, I can not stress enough the quality of this book. This book covers 3D math fundamentals, algorithms, and it is complete with easy to understand (!) proofs. The math is difficult because there is so many problems to be solved in 3D (and they draw from many different branches of mathematics), but it is written in such a clear way that every topic is made approachable. Unlike esoteric Ph. D papers, you aren’t assumed to have any specific knowledge of math idioms or jargon. You simply need a decent grasp of college calculus and trigonometry to make the most of it. There are a few samples too to test your knowledge.



Expect to spend at least a solid year to really make the use of this book. Treat it as you would a two - three semester course in college. During this process, you’ll find yourself occasionally wanting to get more practice and referring to a respective book on it.



If you could only own two books for 3D programming, buy this first and buy Ericson’s book on collision detection next.



In summary, Eric Lengyel’s attention to detail and mastery of 3D math / algorithms really shines and this book is an example of it.

Great book

This book is great for anyone interested in computer graphics. Even for people who do not have a lot of math/graphics experience, this book starts you off with the basics of vectors and matrices and has exercises/solutions for each chapter. It saves you the time of looking through your old linear algebra and differential equation math books and contains the must know information you will use as a graphics programmer.

Math majors rejoice

To be honest, while I find this book to be a decent reference, I find it to be pretty inaccessible in terms of sitting down and reading through it in an attempt to learn the concepts. As a non-math major (I’m actually an engineer and software developer) these math concepts are by no means beyond me. But rather than simply being presented with equation after equation, proof after proof, what I find a lot more valuable is more discussion on the usage of these equations. Specifically I’d like to see examples, diagrams, and code, and there is precious little of any of that in this book.

In other words, this book is very much like what you expect to find in a very dry upper devision college math text for the consumption of math majors who are used to such things. But for a non math major just trying to make use of these concepts in order to get the job done and make games? eh, not so much.

Still, I do think this book is useful as a reference when I want to look up an equation as there are a ton of them crammed into this book, but for me, I just don’t find this book to be very good as a learning tool.

Great book on the math needed for 3D games and graphics

This book illustrates the mathematics that a game programmer would need to develop a professional-quality 3D engine. Although the book is geared toward applications in game development, many of the topics appeal to general interests in 3D graphics. It starts at a fairly basic level in areas such as vector geometry and linear algebra, and then progresses to more advanced topics in 3D game programming such as illumination, visibility determination, and collision detection. Particular attention is given to derivations of key results, ensuring that the reader is not forced to endure gaps in the theory. The book assumes a working knowledge of trigonometry and calculus, but also includes sections that review the important tools used from these disciplines, such as trigonometric identities, differential equations, and Taylor series. This book has plenty of examples and figures, making it much more illustrative than your average math book. I highly recommend it to anyone interested in implementing more advanced mathematics into their games or graphics applications. The following is the table of contents for this second edition:
Chapter 0 The Rendering Pipeline (NEW)
Chapter 1 Vectors
Chapter 2 Matrices
Chapter 3 Transforms
Chapter 4 3D Engine Geometry
Chapter 5 Ray Tracing
Chapter 6 Illumination
Chapter 7 Visibility Determination
Chapter 8 Collision Detection
Chapter 9 Polygonal Techniques
Chapter 10 Shadows (NEW)
Chapter 11 Linear Physics
Chapter 12 Rotational Physics
Chapter 13 Fluid Simulation
Chapter 14 Numerical Methods (NEW)
Chapter 15 Curves and Surfaces (NEW)
Appendix A Complex Numbers
Appendix B Trigonometry Reference
Appendix C Coordinate Systems
Appendix D Taylor Series
Appendix E Answers to Exercises

Note that there are four new chapters in this second edition.

An Essential Source for Robust 3D Graphics Engine Design

I have not yet been able to purchase the second edition of this book. However, since it is a refinement of the first edition, I feel that I am somewhat competent to review this later version of Lengyel’s text on this knowledge of the fundamental chapters which are included in both editions.

Definitely, the subject of 3D Computer Graphics draws programmers and gamers inward like a bug lamp draws insects inward. The ability to express one’s own inspiring representation of the world seems to attract the deepest longings of the subcreative nature of the human person. However, the coder quickly realizes that the realization of this dream is extremely difficult due to the complexities of visualization in nature. Lengyel’s text helps to break through many of these complexities.

In order to make sure that the reader is "up to speed" on the essentials in mathematics which are necessary for a 3D engine, Lengyel gives a quick, but thorough, review of Vector and Matrix operations. I was indeed impressed by the amount that he condensed into the beginning of the text. While this is far from a complete treatise on Linear Algebra, this material gives the reader the basics which will be necessary for all the other chapters. This is further augmented by his treatment of 3D geometry, its representations, and calculations relevant to it.
Using this, he treats on various subjects that are extremely important for making a lean, mean, and beautiful rendering machine.

His treatment of transformations gives the reader a base in this knowledge that is independent of the rendering context. This is an excellent approach because knowledge of how something works allows for one to use it more fully. He also goes further in introducing Quaternions and deriving rotations around any arbitrary axis.

The chapter on Ray Tracing gives the necessary background for the work done in the following chapter on illumination. While the latter is often implemented by the graphics environment that the coder is using, the full knowledge of this allows for further application in subjects such as bump mapping (which he treats) as well as reflective surface simulation.

Perhaps the most interesting and important subject covered in the text, however, is the chapter on visibility determination. Since this is critical to the speed of rendering, this topic is central to game engine design. Lengyel works through the various methods for bounding box construction as well as bounding spheres, ellipsoids, and cylinders and then explains how these can be tested against the view frustrum in order to determine if an object should even be rendered. Another chapter on various techniques which can be applied to polygons outlines procedures for reducing the complexity of meshes, without compromising quality.

The remaining chapters touch on collision detection, linear and rotational physics, and fluid simulation. These topics allow for the addion of various levels of realism to be added to 3D engines and are of importance once the rendering environment has been established. Furthermore, his appendices are a nice addition for those times when our memories fail to recall various facts.

And so, I give this book my definite approval. It is hard to come by a text so compact, yet thorough, on the foundations truly needed for 3D Graphics. These mathematical foundations allow for greater things to be explored and should interest any 3D programmer as well as individuals with a general interest in mathematics, especially in the applications of linear algebra and calculus.