Book Image

Game Physics Cookbook

By : Gabor Szauer
Book Image

Game Physics Cookbook

By: Gabor Szauer

Overview of this book

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!

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Table of Contents (27 chapters)
Game Physics Cookbook
Game Physics Cookbook
Credits
Credits
About the Author
About the Author
Acknowledgements
Acknowledgements
About the Reviewer
About the Reviewer
Acknowledgements
Acknowledgements
www.PacktPub.com
www.PacktPub.com
Customer Feedback
Customer Feedback
Preface
Preface
Free Chapter
1
Vectors
Vectors
Introduction
Vector definition
Component-wise operations
Dot product
Magnitude
Normalizing
Cross product
Angles
Projection
Reflection
2
Matrices
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
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
2D Primitive Shapes
Introduction
2D points
2D lines
Circle
Rectangle
Oriented rectangle
Point containment
Line intersection
5
2D Collisions
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
2D Optimizations
Introduction
Containing circle
Containing rectangle
Simple and complex shapes
Quad tree
Broad phase collisions
7
3D Primitive Shapes
3D Primitive Shapes
Introduction
Point
Line segment
Ray
Sphere
Axis Aligned Bounding Box
Oriented Bounding Box
Plane
Triangle
8
3D Point Tests
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
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
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
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
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
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
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
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
Springs and Joints
Introduction
Particle Modifications
Springs
Cloth
Physics System Modification
Joints
Advanced Topics
Advanced Topics
Introduction
Generic collisions
Stability improvements
Springs
Open source physics engines
Books
Online resources
Summary
Index
Index

Scaling

The scale of a matrix is stored in the main diagonal of the matrix. The scale is stored as a vec3. Each element of the vec3 represents the scale on the corresponding axis. Row and Column major matrices store scale information in the same elements:

The interesting thing with storing scale inside a matrix is that it shares some of the same elements as the rotation part of the matrix. Because of this, extracting the scale of a matrix may not always yield the numbers you would expect.

Getting ready

We're going to implement three functions. One function will retrieve the scale stored inside a matrix. The other two will return a new matrix, containing only the specified scale.

How to do it…

Follow these steps to set and retrieve the scale of a matrix:

  1. Add the declaration for all scaling functions to matrices.h:

    mat4 Scale(float x, float y, float z);
mat4 Scale(const vec3& vec);
vec3 GetScale(const mat4& mat);

  2. Implement the functions that create a 4 X 4 matrix out of scaling information...

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