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Game Physics Cookbook

Game Physics Cookbook

By : Gabor Szauer
4.3 (4)
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Game Physics Cookbook

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!
Table of Contents (19 chapters)
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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
1. Vectors
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1. Vectors
Introduction
Vector definition
Component-wise operations
Dot product
Magnitude
Normalizing
Cross product
Angles
Projection
Reflection
2
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
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
4. 2D Primitive Shapes
4. 2D Primitive Shapes
Introduction
2D points
2D lines
Circle
Rectangle
Oriented rectangle
Point containment
Line intersection
5
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
6. 2D Optimizations
6. 2D Optimizations
Introduction
Containing circle
Containing rectangle
Simple and complex shapes
Quad tree
Broad phase collisions
7
7. 3D Primitive Shapes
7. 3D Primitive Shapes
Introduction
Point
Line segment
Ray
Sphere
Axis Aligned Bounding Box
Oriented Bounding Box
Plane
Triangle
8
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
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
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
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
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
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
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
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
16. Springs and Joints
16. Springs and Joints
Introduction
Particle Modifications
Springs
Cloth
Physics System Modification
Joints
17
A. Advanced Topics
A. Advanced Topics
Introduction
Generic collisions
Stability improvements
Springs
Open source physics engines
Books
Online resources
Summary
18
Index
Index

Reflection

One of the most important concepts in physics for games is collision response and how to react to a collision occurring. More often than not this involves one of the colliding objects bouncing off the other one. We can achieve the bounding through vector reflection. Reflection is also heavily used in many areas of game development, such as graphics programming, to find the color intensity of a fragment.

Given vector Reflection and normal Reflection, we want to find a vector Reflection that is Reflection reflected around Reflection:

Reflection

The reflected vector Reflection can be found with the following formula:

Reflection

Keep in mind, in the preceding equation, Reflection is a unit length vector. This means that theReflection part of the equation actually projects Reflection onto Reflection. If Reflectionwas a non-normalized vector, the preceding equation would be written as follows:

Reflection

Getting ready

Implementing the preceding formula is going to look a little different, this is because we only overloaded the vector scalar multiplication with the scalar being on the right side of the equation. We're going to implement the function assuming Getting ready is already normalized.

How to do it…

Follow these steps to implement a function which will reflect both two and three dimensional vectors.

  1. Add the declaration of the reflection function to vectors.h:
    vec2 Reflection(const vec2& vec, const vec2& normal);
vec3 Reflection(const vec3& vec, const vec3& normal);
  2. Add the implementation of the reflection function to vectors.cpp:
    vec2 Reflection(const vec2& vec,const vec2& normal) {
   float d = Dot(vec, normal);
   return sourceVector - normal * (d * 2.0f );
}

vec3 Reflection(const vec3& vec, const vec3& normal) {
   float d = Dot(vec, normal);
   return sourceVector - normal * (d * 2.0f);
}

How it works…

Given How it works… and How it works…, we're going to find How it works…, which is the reflection of How it works… around How it works…:

How it works…

First, we project How it works… onto How it works…, this operation will yield a vector along How it works… that has the length of How it works…:

How it works…

We want to find the reflected vector How it works…. The following figure shows How it works… in two places, remember it doesn't matter where you draw a vector as long as its components are the same:

How it works…

Looking at the preceding figure, we can tell that subtracting How it works… from How it works… will result in How it works…:

How it works…

This is how we get to the final formula, How it works….

End of Chapter 1
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