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Home > Game Development > Game Design > Game Physics Cookbook
Game Physics Cookbook
Game Physics Cookbook

Game Physics Cookbook: Discover over 100 easy-to-follow recipes to help you implement efficient game physics and collision detection in your games

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Game Physics Cookbook

Chapter 2. Matrices

In this chapter, we will cover the basic math needed to multiply and invert matrices:

  • Definition
  • Transpose
  • Multiplication
  • Identity matrix
  • Determinant of a 2x2 matrix
  • Matrix of minors
  • Matrix of cofactors
  • Determinant of a 3x3 matrix
  • Operations of a 4x4 matrix
  • Adjugate matrix
  • Matrix inverse

Introduction

Matrices in games are used extensively. In the context of physics, matrices are used to represent different coordinate spaces. In games, we often combine coordinate spaces; this is done through matrix multiplication. In game physics, it's useful to move one object into the coordinate space of another object; this requires matrices to be inverted. In order to invert a matrix, we have to find its minor, determinant, cofactor, and adjugate. This chapter focuses on what is needed to multiply and invert matrices.

Note

Every formula in this chapter is followed by some practical examples. If you find yourself needing additional examples, Purplemath is a great resource; look under the Advanced Algebra Topic section: www.purplemath.com/modules

Matrix definition

A matrix is a Matrix definition grid of numbers, represented by a bold capital letter. The number of rows in a matrix is represented by i; the number of columns is represented by j.

For example, in a 3 X 2 matrix, i would be 3 and j would be 2. This 3 X 2 matrix looks like this:

Matrix definition

Matrices can be of any dimension; in video games, we tend to use 2 X 2, 3 X 3, and 4 X 4 matrices. If a matrix has the same number of rows and columns, it is called a square matrix. In this book, we're going to be working mostly with square matrices.

Individual elements of the matrix are indexed with subscripts. For example, Matrix definitionrefers to the element in row 1, column 2 of the matrix M.

Getting ready

We are going to implement a 2 X 2, 3 X 3, and 4 X 4 matrix. Internally, each matrix will be represented as a linear array of memory. Much like vectors, we will use an anonymous union to support a variety of access patterns. Pay attention to how the indexing operator is overridden, matrix indices in code start at 0, not...

Transpose

The transpose of matrix M, written as Transpose is a matrix in which every element i, j equals the element j, i of the original matrix. The transpose of a matrix can be acquired by reflecting the matrix over its main diagonal, writing the rows of M as the columns of Transpose, or by writing the columns of M as the rows of Transpose. We can express the transpose for each component of a matrix with the following equation:

Transpose

The transpose operation replaces the rows of a matrix with its columns:

Transpose Transpose

Getting ready

We're going to create a non-nested loop that serves as a generic Transpose function. This function will be able to transpose matrices of any dimension. We're then going to create Transpose functions specific to 2 X 2, 3 X 3, and 4 X 4 matrices. These more specific functions are going to call the generic Transpose with the appropriate arguments.

How to do it…

Follow these steps to implement a generic transpose function and transpose functions for two, three and four dimensional square...

Multiplication

Like a vector, there are many ways to multiply a matrix. In this chapter we will cover multiplying matrices by a scalar or by another matrix. Scalar multiplication is component wise. Given a Multiplication matrix M and a scalar s, scalar multiplication is defined as follows:

Multiplication

We can also multiply a matrix by another matrix. Two matrices, A and B, can be multiplied together only if the number of columns in A matches the number of rows in B. That is, two matrices can only be multiplied together if their inner dimensions match.

When multiplying two matrices together, the dimension of the resulting matrix will match the outer dimensions of the matrices being multiplied. If A is an Multiplication matrix and B is an Multiplication matrix, the product of AB will be an Multiplication matrix. We can find each element of the matrix AB with the following formula:

Multiplication

This operation concatenates the transformations represented by the two matrices into one matrix. Matrix multiplication is not cumulative. Multiplication. However, matrix multiplication is associative...

Identity matrix

Multiplying a scalar number by 1 will result in the original scalar number. There is a matrix analogue to this, the identity matrix. The identity matrix is commonly written as I. If a matrix is multiplied by the identity matrix, the result is the original matrix Identity matrix.

In the identity matrix, all non-diagonal elements are 0, while all diagonal elements are one Identity matrix. The identity matrix looks like this:

Identity matrix

Getting ready

Because the identity matrix has no effect on multiplication, by convention it is the default value for all matrices. We're going to add two constructors to every matrix struct. One of the constructors is going to take no arguments; this will create an identity matrix. The other constructor will take one float for every element of the matrix and assign every element inside the matrix. Both constructors are going to be inline.

How to do it…

Follow these steps to add both a default and overloaded constructors to matrices:

  1. Add the default inline constructor to the mat2...

Determinant of a 2x2 matrix

Determinants are useful for solving systems of linear equations; however, in the context of a 3D physics engine, we use them almost exclusively to find the inverse of a matrix. The determinant of a matrix M is a scalar value, it's denoted as Determinant of a 2x2 matrix.The determinant of a matrix is the same as the determinant of its transpose Determinant of a 2x2 matrix.

We can use a shortcut to find the determinant of a 2 X 2 matrix; subtract the product of the diagonals. This is actually the manually expanded form of Laplace Expansion; we will cover the proper formula in detail later:

Determinant of a 2x2 matrix

One interesting property of determinants is that the determinant of the inverse of a matrix is the same as the inverse determinant of that matrix:

Determinant of a 2x2 matrix

Finding the determinant of a 2 X 2 matrix is fairly straightforward, as we have already expanded the formula. We're just going to implement this in code.

How to do it…

Follow these steps to implement a function which returns the determinant of a 2 X 2 matrix:

  1. Add the declaration...

Introduction

Matrices in games are used extensively. In the context of physics, matrices are used to represent different coordinate spaces. In games, we often combine coordinate spaces; this is done through matrix multiplication. In game physics, it's useful to move one object into the coordinate space of another object; this requires matrices to be inverted. In order to invert a matrix, we have to find its minor, determinant, cofactor, and adjugate. This chapter focuses on what is needed to multiply and invert matrices.

Note

Every formula in this chapter is followed by some practical examples. If you find yourself needing additional examples, Purplemath is a great resource; look under the Advanced Algebra Topic section: www.purplemath.com/modules

Matrix definition

A matrix is a grid of numbers, represented by a bold capital letter. The number of rows in a matrix is represented by i; the number of columns is represented by j.

For example, in a 3 X 2 matrix, i would be 3 and j would be 2. This 3 X 2 matrix looks like this:

Matrices can be of any dimension; in video games, we tend to use 2 X 2, 3 X 3, and 4 X 4 matrices. If a matrix has the same number of rows and columns, it is called a square matrix. In this book, we're going to be working mostly with square matrices.

Individual elements of the matrix are indexed with subscripts. For example, refers to the element in row 1, column 2 of the matrix M.

Getting ready

We are going to implement a 2 X 2, 3 X 3, and 4 X 4 matrix. Internally, each matrix will be represented as a linear array of memory. Much like vectors, we will use an anonymous union to support a variety of access patterns. Pay attention to how the indexing operator is overridden, matrix indices in code start at 0, not 1. This...

Transpose

The transpose of matrix M, written as is a matrix in which every element i, j equals the element j, i of the original matrix. The transpose of a matrix can be acquired by reflecting the matrix over its main diagonal, writing the rows of M as the columns of , or by writing the columns of M as the rows of . We can express the transpose for each component of a matrix with the following equation:

The transpose operation replaces the rows of a matrix with its columns:

Getting ready

We're going to create a non-nested loop that serves as a generic Transpose function. This function will be able to transpose matrices of any dimension. We're then going to create Transpose functions specific to 2 X 2, 3 X 3, and 4 X 4 matrices. These more specific functions are going to call the generic Transpose with the appropriate arguments.

How to do it…

Follow these steps to implement a generic transpose function and transpose functions for two, three and four dimensional square matrices:

  1. Add...

Multiplication

Like a vector, there are many ways to multiply a matrix. In this chapter we will cover multiplying matrices by a scalar or by another matrix. Scalar multiplication is component wise. Given a matrix M and a scalar s, scalar multiplication is defined as follows:

We can also multiply a matrix by another matrix. Two matrices, A and B, can be multiplied together only if the number of columns in A matches the number of rows in B. That is, two matrices can only be multiplied together if their inner dimensions match.

When multiplying two matrices together, the dimension of the resulting matrix will match the outer dimensions of the matrices being multiplied. If A is an matrix and B is an matrix, the product of AB will be an matrix. We can find each element of the matrix AB with the following formula:

This operation concatenates the transformations represented by the two matrices into one matrix. Matrix multiplication is not cumulative. . However, matrix multiplication is associative...

Identity matrix

Multiplying a scalar number by 1 will result in the original scalar number. There is a matrix analogue to this, the identity matrix. The identity matrix is commonly written as I. If a matrix is multiplied by the identity matrix, the result is the original matrix .

In the identity matrix, all non-diagonal elements are 0, while all diagonal elements are one . The identity matrix looks like this:

Getting ready

Because the identity matrix has no effect on multiplication, by convention it is the default value for all matrices. We're going to add two constructors to every matrix struct. One of the constructors is going to take no arguments; this will create an identity matrix. The other constructor will take one float for every element of the matrix and assign every element inside the matrix. Both constructors are going to be inline.

How to do it…

Follow these steps to add both a default and overloaded constructors to matrices:

  1. Add the default inline constructor to the mat2 struct:

    inline...

Determinant of a 2x2 matrix

Determinants are useful for solving systems of linear equations; however, in the context of a 3D physics engine, we use them almost exclusively to find the inverse of a matrix. The determinant of a matrix M is a scalar value, it's denoted as .The determinant of a matrix is the same as the determinant of its transpose .

We can use a shortcut to find the determinant of a 2 X 2 matrix; subtract the product of the diagonals. This is actually the manually expanded form of Laplace Expansion; we will cover the proper formula in detail later:

One interesting property of determinants is that the determinant of the inverse of a matrix is the same as the inverse determinant of that matrix:

Finding the determinant of a 2 X 2 matrix is fairly straightforward, as we have already expanded the formula. We're just going to implement this in code.

How to do it…

Follow these steps to implement a function which returns the determinant of a 2 X 2 matrix:

  1. Add the declaration for the determinant...

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Key benefits

  • Get a comprehensive coverage of techniques to create high performance collision detection in games
  • Learn the core mathematics concepts and physics involved in depicting collision detection for your games
  • Get a hands-on experience of building a rigid body physics engine

Description

Physics is really important for game programmers who want to add realism and functionality to their games. Collision detection in particular is a problem that affects all game developers, regardless of the platform, engine, or toolkit they use. This book will teach you the concepts and formulas behind collision detection. You will also be taught how to build a simple physics engine, where Rigid Body physics is the main focus, and learn about intersection algorithms for primitive shapes. You’ll begin by building a strong foundation in mathematics that will be used throughout the book. We’ll guide you through implementing 2D and 3D primitives and show you how to perform effective collision tests for them. We then pivot to one of the harder areas of game development—collision detection and resolution. Further on, you will learn what a Physics engine is, how to set up a game window, and how to implement rendering. We’ll explore advanced physics topics such as constraint solving. You’ll also find out how to implement a rudimentary physics engine, which you can use to build an Angry Birds type of game or a more advanced game. By the end of the book, you will have implemented all primitive and some advanced collision tests, and you will be able to read on geometry and linear Algebra formulas to take forward to your own games!

Who is this book for?

This book is for beginner to intermediate game developers. You don’t need to have a formal education in games—you can be a hobbyist or indie developer who started making games with Unity 3D.

What you will learn

  • Implement fundamental maths so you can develop solid game physics
  • Use matrices to encode linear transformations
  • Know how to check geometric primitives for collisions
  • Build a Physics engine that can create realistic rigid body behavior
  • Understand advanced techniques, including the Separating Axis Theorem
  • Create physically accurate collision reactions
  • Explore spatial partitioning as an acceleration structure for collisions
  • Resolve rigid body collisions between primitive shapes
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Publication date : Mar 24, 2017
Length: 480 pages
Edition : 1st
Language : English
ISBN-13 : 9781787123663
Vendor :
Unity Technologies
Category :
Game Development
Languages :
C++
Concepts :
Game Design
Tools :
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Publication date : Mar 24, 2017
Length: 480 pages
Edition : 1st
Language : English
ISBN-13 : 9781787123663
Vendor :
Unity Technologies
Category :
Game Development
Languages :
C++
Concepts :
Game Design
Tools :
Unity

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Table of Contents

18 Chapters
1. Vectors Chevron down icon Chevron up icon
1. Vectors
Introduction
Vector definition
Component-wise operations
Dot product
Magnitude
Normalizing
Cross product
Angles
Projection
Reflection
2. Matrices Chevron down icon Chevron up icon
2. Matrices
Introduction
Matrix definition
Transpose
Multiplication
Identity matrix
Determinant of a 2x2 matrix
Matrix of minors
Cofactor
Determinant of a 3x3 matrix
Operations on a 4x4 matrix
Adjugate matrix
Matrix inverse
3. Matrix Transformations Chevron down icon Chevron up icon
3. Matrix Transformations
Introduction
Matrix majors
Translation
Scaling
How rotations work
Rotation matrices
Axis angle rotation
Vector matrix multiplication
Transform matrix
View matrix
Projection matrix
4. 2D Primitive Shapes Chevron down icon Chevron up icon
4. 2D Primitive Shapes
Introduction
2D points
2D lines
Circle
Rectangle
Oriented rectangle
Point containment
Line intersection
5. 2D Collisions Chevron down icon Chevron up icon
5. 2D Collisions
Introduction
Circle to circle
Circle to rectangle
Circle to oriented rectangle
Rectangle to rectangle
Separating Axis Theorem
Rectangle to oriented rectangle
Oriented rectangle to oriented rectangle
6. 2D Optimizations Chevron down icon Chevron up icon
6. 2D Optimizations
Introduction
Containing circle
Containing rectangle
Simple and complex shapes
Quad tree
Broad phase collisions
7. 3D Primitive Shapes Chevron down icon Chevron up icon
7. 3D Primitive Shapes
Introduction
Point
Line segment
Ray
Sphere
Axis Aligned Bounding Box
Oriented Bounding Box
Plane
Triangle
8. 3D Point Tests Chevron down icon Chevron up icon
8. 3D Point Tests
Introduction
Point and sphere
Point and AABB
Point and Oriented Bounding Box
Point and plane
Point and line
Point and ray
9. 3D Shape Intersections Chevron down icon Chevron up icon
9. 3D Shape Intersections
Introduction
Sphere-to-sphere
Sphere-to-AABB
Sphere-to-OBB
Sphere-to-plane
AABB-to-AABB
AABB-to-OBB
AABB-to-plane
OBB-to-OBB
OBB-to-plane
Plane-to-plane
10. 3D Line Intersections Chevron down icon Chevron up icon
10. 3D Line Intersections
Introduction
Raycast Sphere
Raycast Axis Aligned Bounding Box
Raycast Oriented Bounding Box
Raycast plane
Linetest Sphere
Linetest Axis Aligned Bounding Box
Linetest Oriented Bounding Box
Linetest Plane
11. Triangles and Meshes Chevron down icon Chevron up icon
11. Triangles and Meshes
Introduction
Point in triangle
Closest point triangle
Triangle to sphere
Triangle to Axis Aligned Bounding Box
Triangle to Oriented Bounding Box
Triangle to plane
Triangle to triangle
Robustness of the Separating Axis Theorem
Raycast Triangle
Linetest Triangle
Mesh object
Mesh optimization
Mesh operations
12. Models and Scenes Chevron down icon Chevron up icon
12. Models and Scenes
Introduction
The Model object
Operations on models
The Scene object
Operations on the scene
The Octree object
Octree contents
Operations on the Octree
Octree scene integration
13. Camera and Frustum Chevron down icon Chevron up icon
13. Camera and Frustum
Introduction
Camera object
Camera controls
Frustum object
Frustum from matrix
Sphere in frustum
Bounding Box in frustum
Octree culling
Picking
14. Constraint Solving Chevron down icon Chevron up icon
14. Constraint Solving
Introduction
Framework introduction
Raycast sphere
Raycast Bounding Box
Raycast plane and triangle
Physics system
Integrating particles
Solving constraints
Verlet Integration
15. Manifolds and Impulses Chevron down icon Chevron up icon
15. Manifolds and Impulses
Introduction
Manifold for spheres
Manifold for boxes
Rigidbody Modifications
Linear Velocity
Linear Impulse
Physics System Update
Angular Velocity
Angular Impulse
16. Springs and Joints Chevron down icon Chevron up icon
16. Springs and Joints
Introduction
Particle Modifications
Springs
Cloth
Physics System Modification
Joints
A. Advanced Topics Chevron down icon Chevron up icon
A. Advanced Topics
Introduction
Generic collisions
Stability improvements
Springs
Open source physics engines
Books
Online resources
Summary
Index Chevron down icon Chevron up icon
Index

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Customer reviews

Rating distribution
Full star icon Full star icon Full star icon Full star icon Half star icon 4.3
(4 Ratings)
5 star 50%
4 star 25%
3 star 25%
2 star 0%
1 star 0%
Billy Feb 26, 2019
Full star icon Full star icon Full star icon Full star icon Full star icon 5
Good book.Now I can build my flight simulator
Amazon Verified review Amazon
Jonathan Jul 22, 2017
Full star icon Full star icon Full star icon Full star icon Full star icon 5
This is is a great book. They say you can’t judge a book by its cover. I say you can judge it by its source code. I bought two other game physics books and this is the only one I successfully converted to C for Mac while reading through it. I searched Google for game physics source code and found this book’s source code on GitHub. I learned the hard way that if you can’t download the source code and the code in the book is not complete don’t buy it! The only thing I would change is to get rid of the 2D chapters and replace them with terrain and plane collision response recipes later on. Also, there are a ton of typos in the recipe descriptions, but the book’s formulas and code worked fine. And I always had the GiHub source code download to refer back to. Note that in the book it mostly explains how to get the physics working and not the OpenGL graphics. However, the GitHub download should have working OpenGL code. If you are looking for a good game physics book, Game Physics Cookbook is the one to buy.
Amazon Verified review Amazon
Persnickety Sep 26, 2017
Full star icon Full star icon Full star icon Full star icon Empty star icon 4
If you are new to computer physics programming then this is a must have book. It keeps things simple and direct. Explanations are well done and the math is kept simple. I combined this with the "real time collision detection" book and feel like I have a good grounding in the 3d game physics area. Both books tend to focus on basic shapes and their interactions but also give plenty of good info on using those basic shapes to build more sophisticated physical interactions. Where this books falls short is in the 2D area, and this is the reason I took away 1/2 star. In one place the 2D code uses a point slope form of the line to compute an intersection and introduces a divide by zero error directly in the code. The other 1/2 star is taken away due to numerous coding errors and inconsistencies throughout the text. Also the errata for the book is practically nil.But don't let that put you off: the book is an excellent intro to physics programming in general.
Amazon Verified review Amazon
John Hague Dec 14, 2020
Full star icon Full star icon Full star icon Empty star icon Empty star icon 3
Whilst this book provides a useful revision for game industry techniques, this edition cannot be recommended to new starters in the field. This is because this edition is packed with typos and mistakes in both text and code snippets. You must use the GitHub code in conjunction to review and iron out these mistakes: which is a disappointing lack of attention to detail before the book went to press.
Amazon Verified review Amazon

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About the author

Profile icon Gabor Szauer
Gabor Szauer
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Gabor Szauer has been making games since 2010. He graduated from Full Sail University in 2010 with a bachelor's degree in game development. Gabor maintains an active Twitter presence, and maintains a programming-oriented game development blog. Gabor's previously published books are Game Physics Programming Cookbook and Lua Quick Start Guide, both published by Packt.
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