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Game Physics Cookbook

By : Gabor Szauer

4.3 (4)

Game Physics Cookbook

4.3 (4)

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!

Preface

Preface

What this book covers

What you need for this book

Who this book is for

Sections

Conventions

Reader feedback

Customer support

Free Chapter

1. Vectors

1. Vectors

Introduction

Vector definition

Component-wise operations

Dot product

Magnitude

Normalizing

Cross product

Angles

Projection

Reflection

2. Matrices

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

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

4. 2D Primitive Shapes

Introduction

2D points

2D lines

Circle

Rectangle

Oriented rectangle

Point containment

Line intersection

5. 2D Collisions

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

6. 2D Optimizations

Introduction

Containing circle

Containing rectangle

Simple and complex shapes

Quad tree

Broad phase collisions

7. 3D Primitive Shapes

7. 3D Primitive Shapes

Introduction

Point

Line segment

Ray

Sphere

Axis Aligned Bounding Box

Oriented Bounding Box

Plane

Triangle

8. 3D Point Tests

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

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

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

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

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

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

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

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

16. Springs and Joints

Introduction

Particle Modifications

Springs

Cloth

Physics System Modification

Joints

A. Advanced Topics

A. Advanced Topics

Introduction

Generic collisions

Stability improvements

Springs

Open source physics engines

Books

Online resources

Summary

Index

Index

Customer Reviews

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Normalizing

A vector with a magnitude of 1 is a normal vector, sometimes called a unit vector. Whenever a vector has a length of 1, we can say that it has unit length. A normal vector is written as the letter of the vector with a caret symbol on top instead of an arrow, Normalizing . We can normalize any vector by dividing each of its components by the length of the vector:

Normalizing

We never implemented division operators for the vector class. We can rewrite the preceding equation as reciprocal multiplication. This means we can obtain the normal of a vector if we multiply that vector by the inverse of its length:

Normalizing

Getting ready

We are going to implement two functions, Normalize and Normalized. The first function will change the input vector to have a length of 1. The second function will not change the input vector; rather it will return a new vector with a length of 1.

How to do it…

Follow these steps to implement functions which will make a vector unit length or return a unit length vector. These steps utilize reciprocal multiplication.

Declare the Normalize and Normalized functions in vectors.h:

void Normalize(vec2& v);
void Normalize(vec3& v);

vec2 Normalized(const vec2& v);
vec3 Normalized(const vec3& v);

Add the implementation of these functions to vectors.cpp:

void Normalize(vec2& v) {
   v = v * (1.0f / Magnitude(v));
}

void Normalize(vec3& v) {
   v = v * (1.0f / Magnitude(v));
}

vec2 Normalized(const vec2& v) {
   return v * (1.0f / Magnitude(v));
}

vec3 Normalized(const vec3& v) {
   return v * (1.0f / Magnitude(v));
}

How it works…

Normalizing works by scaling the vector by the inverse of its length. This scale makes the vector have unit length, which is a length of 1. Unit vectors are special as any number multiplied by 1 stays the same number. This makes unit vectors ideal for representing a direction. If a direction has unit length, scaling it by some velocity becomes trivial.

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