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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

Angles

We have had a brief introduction to the angle between vectors when we discussed the dot product and the magnitude of a vector. In this recipe, we will discuss how to find the actual angle between two vectors. The formula to find angle theta between two vectors is:

Angles

Getting ready

We have already implemented both the dot product and magnitude functions for vectors; this means we have everything needed to find the angle between two vectors already written. In general, this is a very expensive function, as it performs two square roots and an inverse cosine. Because it's such an expensive function, we try to avoid it whenever possible.

We can save a little bit of performance if, instead of multiplying the length of both vectors, we multiply the squared length of the vectors and then do just one square root operation on the result.

How to do it…

  1. Add the declaration of the angle function to vectors.h:
    float Angle(const vec2& l, const vec2& r);
float Angle(const vec3& l, const vec3& r);
  2. Provide the implementation of the angle function in vectors.cpp:
    float Angle(const vec2& l, const vec2& r) {
   float m = sqrtf(MagnitudeSq(l) * MagnitudeSq(r));
   return acos(Dot(l, r) / m);
}

float Angle(const vec3& l, const vec3& r) {
   float m = sqrtf(MagnitudeSq(l) * MagnitudeSq(r));
   return acos(Dot(l, r) / m);
}

How it works…

This formula relies on the geometric definition of the dot product:

How it works…

This formula states that the dot product of two vectors is the cosine of the angle between them multiplied by both of their lengths. We can rewrite this formula with the cosine being isolated if we divide both sides by the product of the lengths of How it works… and How it works…:

How it works…

We can now use the inverse of cosine, the arc cosine (acos), to find the angle theta:

How it works…

There's more…

The acos function we used to find the angle between vectors comes from the standard C math library. This implementation of acos returns radians, not degrees. It's much more intuitive to think of angles in terms of degrees than radians.

Radians and degrees

Add the following macros to the top of the vectors.h header file:

#define RAD2DEG(x) ((x) * 57.295754f)
#define DEG2RAD(x) ((x) * 0.0174533f)

Using these macros you can convert between radians and degrees. For example, if you wanted to get the angle in degrees between vectors Radians and degrees and Radians and degrees, you could use the following code:

float degrees = RAD2DEG(Angle(A, B));

If you are interested in the math used to derive these numbers, I suggest watching the following Khan Academy video:

https://www.khanacademy.org/math/algebra2/trig-functions/intro-to-radians-alg2/v/introduction-to-radians

End of Section 9
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