Mathematics
Richard Hammock,
Book of Proof.
It is not uncommon for computer science majors to finish their undergraduate degrees with little experience proving mathematical statements in a formal manner. Book of Proof is a very gentle introduction to proof techniques that has filled in a lot of the gaps in my own education. I bought
How to Prove It: A Structured Approach
at the same time and found Book of Proof the much more useful of the two. After a short review of logic and set theory it takes you through the most common proof patterns, with plenty of well-paced examples. Highly recommended, and also available as a free
ebook.
Sheldon Axler,
Linear Algebra Done Right.
After years of computer graphics / engineering focused linear algebra this was my first exposure to a more mathematical treatment. The book emphasizes linear maps, rather than their representation (matrices) which gave me a new appreciation for the subject, and helped me see the generalizations and abstractions behind linear algebra. Part of me wishes this was my first exposure to the subject, but in any case, if you've come from a Strang style linear algebra course then you should get a lot out of it.
Gene H. Golub, Charles F. Van Loan,
Matrix Computations
More of a reference manual than a text book, this has everything you need to know to implement advanced linear algebra methods. Although it is rather dense, it shows how to look at certain operations from a number of different perspectives. For example, I found the description of right-looking LU decomposition really gave me a new intuition for the process that I hadn't seen elsewhere.
Justin Solomon,
Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics.
This text grew out of the author's Stanford course on numerical methods (
CS205). The book covers a lot of ground, but the theme running throughout is on the variational view of problems, which I personally find quite appealing. However, if you want a deeper look at optimization methods specifically, then I would recommend
Numerical Optimization
by Nocedal & Wright which is also surprisingly nice to read.
Computer Graphics
Matt Pharr, Wenzel Jakob, Greg Humphreys,
Physically Based Rendering.
Widely known as the best reference on ray-tracing and light transport, and rightly so. This book really is one of the greats. I lugged it with me on the London underground every day for a year to read it on my commute, and implemented it at night when I got home. The chapters on sampling theory and volume rendering are great, but even the small implementation details like generating viewing transforms without ever calculating an inverse are neat takeaways. You can see some images I made while implementing some of the ideas in this book
here.
Physics
Ahmed A. Shabana,
Computational Dynamics.
One of the few books that goes into the details of not only the physical formulation but also the numerical methods to solve them. This is a book aimed at mechanical engineers, not game programmers. It covers 2D (planar), and 3D (spatial) rigid body dynamics and derives all the common joint types. It discusses how to set up the Newton-Euler equations of motion, and solve them using a variety of direct methods (as opposed to iterative methods usually employed in real-time). One thing it does not cover is contact, or unilateral constraints. Nevertheless, I feel like this is a rare gem that gets the theory right, and at the same time gives plenty of implementation detail to make it concrete.
Ahmed A. Shabana,
Dynamics of Multibody Systems.
This is a follow up work to Computational Dynamics, but with a focus on formulation rather than computation. In fact there is hardly a mention anywhere in the book of how to solve the resulting equations, but a much deeper development of the rigid body equations of motion in generalized coordinates through the Euler-Lagrange equation. Covers many different rotation parameterizations, but with a focus on quaternions. The sections on deformable bodies are perhaps not very relevant for graphics applications, but if you want a deeper understanding of rotation parameterizations, and their relation to rigid body dynamics then this is the best treatment I've seen.
Cornelius Lanczos,
The Variational Principles of Mechanics.
This was my first introduction the calculus of variations, and the principle of least-action. The exposition is clear, and straightforward, without assuming a great mathematical background. This is a short book that moves quite fast, I recommend pairing it with
Leonard Susskind's lectures on Classical Mechanics.
Goldstien, Safko, Poole,
Classical Mechanics.
I obtained a well-used copy of the third edition of this highly cited text on classical mechanics. Despite being widely recommended, Goldstein's book has been probably the least useful for my own development. I struggled with notation and lack of examples, and found Shabana's books to be more relevant to the end goal of writing simulation software.
Robert Bridson,
Fluid Simulation for Computer Graphics.
Just as the title says, this is easily the best reference for fluid simulation in computer graphics. Focus is primarily on grid-based methods, and they are developed in-depth with enough implementation detail to make it quite possible to write your own high-quality fluid simulator from start to finish while reading through. The second edition includes a short chapter on vortex based formulations, however this is not yet as complete as the rest of the book (it is still an active area of research). Some of the content is also available as part of the course notes for the SIGGRAPH 2007 course on fluid simulation, but just get the book, you'll have a lot of fun.
Christer Ericson,
Real-Time Collision Detection.
If you don't already know this book already then you have been missing out. Pretty much the de-facto standard collision detection reference. Not only does it have all the common intersection and overlap tests, it covers acceleration structures, computer architecture, and optimization. Everything I know about collision detection I learned from this book, and much more.
Gino van den Bergen and Dirk Gregorius (editors),
Game Physics Pearls.
This is the physics equivalent of a Graphics Gems book. The quality of the articles in here is particularly good, with highlights being the quaternion constraints article by Claude Lacoursière, and the chapter on debugging physics simulations. One of the mercifully sized books you can carry on a plane and just pick up to read a chapter at leisure. It would be great to see this become a fully fledged series.
Richard Hammock, Book of Proof.
It is not uncommon for computer science majors to finish their undergraduate degrees with little experience proving mathematical statements in a formal manner. Book of Proof is a very gentle introduction to proof techniques that has filled in a lot of the gaps in my own education. I bought How to Prove It: A Structured Approach
at the same time and found Book of Proof the much more useful of the two. After a short review of logic and set theory it takes you through the most common proof patterns, with plenty of well-paced examples. Highly recommended, and also available as a free ebook.
Sheldon Axler, Linear Algebra Done Right.
After years of computer graphics / engineering focused linear algebra this was my first exposure to a more mathematical treatment. The book emphasizes linear maps, rather than their representation (matrices) which gave me a new appreciation for the subject, and helped me see the generalizations and abstractions behind linear algebra. Part of me wishes this was my first exposure to the subject, but in any case, if you've come from a Strang style linear algebra course then you should get a lot out of it.
Gene H. Golub, Charles F. Van Loan, Matrix Computations
More of a reference manual than a text book, this has everything you need to know to implement advanced linear algebra methods. Although it is rather dense, it shows how to look at certain operations from a number of different perspectives. For example, I found the description of right-looking LU decomposition really gave me a new intuition for the process that I hadn't seen elsewhere.
Justin Solomon, Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics.
This text grew out of the author's Stanford course on numerical methods (CS205). The book covers a lot of ground, but the theme running throughout is on the variational view of problems, which I personally find quite appealing. However, if you want a deeper look at optimization methods specifically, then I would recommend Numerical Optimization
by Nocedal & Wright which is also surprisingly nice to read.
Matt Pharr, Wenzel Jakob, Greg Humphreys, Physically Based Rendering.
Widely known as the best reference on ray-tracing and light transport, and rightly so. This book really is one of the greats. I lugged it with me on the London underground every day for a year to read it on my commute, and implemented it at night when I got home. The chapters on sampling theory and volume rendering are great, but even the small implementation details like generating viewing transforms without ever calculating an inverse are neat takeaways. You can see some images I made while implementing some of the ideas in this book here.
Ahmed A. Shabana, Computational Dynamics.
One of the few books that goes into the details of not only the physical formulation but also the numerical methods to solve them. This is a book aimed at mechanical engineers, not game programmers. It covers 2D (planar), and 3D (spatial) rigid body dynamics and derives all the common joint types. It discusses how to set up the Newton-Euler equations of motion, and solve them using a variety of direct methods (as opposed to iterative methods usually employed in real-time). One thing it does not cover is contact, or unilateral constraints. Nevertheless, I feel like this is a rare gem that gets the theory right, and at the same time gives plenty of implementation detail to make it concrete.
Ahmed A. Shabana, Dynamics of Multibody Systems.
This is a follow up work to Computational Dynamics, but with a focus on formulation rather than computation. In fact there is hardly a mention anywhere in the book of how to solve the resulting equations, but a much deeper development of the rigid body equations of motion in generalized coordinates through the Euler-Lagrange equation. Covers many different rotation parameterizations, but with a focus on quaternions. The sections on deformable bodies are perhaps not very relevant for graphics applications, but if you want a deeper understanding of rotation parameterizations, and their relation to rigid body dynamics then this is the best treatment I've seen.
Cornelius Lanczos, The Variational Principles of Mechanics.
This was my first introduction the calculus of variations, and the principle of least-action. The exposition is clear, and straightforward, without assuming a great mathematical background. This is a short book that moves quite fast, I recommend pairing it with Leonard Susskind's lectures on Classical Mechanics.
Goldstien, Safko, Poole, Classical Mechanics.
I obtained a well-used copy of the third edition of this highly cited text on classical mechanics. Despite being widely recommended, Goldstein's book has been probably the least useful for my own development. I struggled with notation and lack of examples, and found Shabana's books to be more relevant to the end goal of writing simulation software.
Robert Bridson, Fluid Simulation for Computer Graphics.
Just as the title says, this is easily the best reference for fluid simulation in computer graphics. Focus is primarily on grid-based methods, and they are developed in-depth with enough implementation detail to make it quite possible to write your own high-quality fluid simulator from start to finish while reading through. The second edition includes a short chapter on vortex based formulations, however this is not yet as complete as the rest of the book (it is still an active area of research). Some of the content is also available as part of the course notes for the SIGGRAPH 2007 course on fluid simulation, but just get the book, you'll have a lot of fun.
Christer Ericson, Real-Time Collision Detection.
If you don't already know this book already then you have been missing out. Pretty much the de-facto standard collision detection reference. Not only does it have all the common intersection and overlap tests, it covers acceleration structures, computer architecture, and optimization. Everything I know about collision detection I learned from this book, and much more.
Gino van den Bergen and Dirk Gregorius (editors), Game Physics Pearls.
This is the physics equivalent of a Graphics Gems book. The quality of the articles in here is particularly good, with highlights being the quaternion constraints article by Claude Lacoursière, and the chapter on debugging physics simulations. One of the mercifully sized books you can carry on a plane and just pick up to read a chapter at leisure. It would be great to see this become a fully fledged series.