Browse Library
Sign In Start Free Trial
Close icon Search icon
Account
Browse Library Advanced Search Sign In Start Free Trial

Add to playlist

Create a Playlist

Modal Close icon
You need to login to use this feature.
  • Game Physics Cookbook
  • Toc
  • feedback
Game Physics Cookbook

Game Physics Cookbook

By : Gabor Szauer
4.3 (4)
close
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)
close
close
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
chevron up
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

Dot product

The dot product, sometimes referred to as scalar product or inner product between two vectors, returns a scalar value. It's written as a dot between two vectors, Dot product. The formula for the dot product is defined as follows:

Dot product

The sigma symbol Dot product means sum (add) everything up that follows. The number on top of the sigma is the upper limit; the variable on the bottom is the lower limit. If n and i is 0, the subscripts 0, 1, and 2 are processed. Without using the sigma symbol, the preceding equation would look like this:

Dot product

The resulting scalar represents the directional relation of the vectors. That is, Dot product represents how much Dot product is pointing in the direction of Dot product. Using the dot product we can tell if two vectors are pointing in the same direction or not following these rules:

  • If the dot product is positive, the vectors are pointing in the same direction
  • If the dot product is negative, the vectors point in opposing directions
  • If the dot product is 0, the vectors are perpendicular

How to do it…

Follow these steps to implement the dot product for two and three dimensional vectors:

  1. Add the declaration for the dot product to vectors.h:
    float Dot(const vec2& l, const vec2& r);
float Dot(const vec3& l, const vec3& r);
  2. Add the implementation for the dot product to vector.cpp:
    float Dot(const vec2& l, const vec2& r) {
   return l.x * r.x + l.y * r.y;
}

float Dot(const vec3& l, const vec3& r) {
   return l.x * r.x + l.y * r.y + l.z * r.z;
}

How it works…

Given the formula and the code for the dot product, let's see an example of what we could use it for. Assume we have a spaceship S. We know its forward vector, How it works… and a vector that points to its right, How it works…:

How it works…

We also have an enemy ship E, and a vector that points from our ship S to the enemy ship E, vector How it works…:

How it works…

How can we tell if the the ship S needs to turn left or right to face the enemy ship E?

We need to take the dot product of How it works… and How it works…. If the result of the dot product is positive, the ship needs to turn right. If the result of the dot product is negative, the ship needs to turn to the left. If the result of the dot product is 0, the ship does not need to turn.

There's more…

Our definition of the dot product is fairly abstract. We know that the dot product gives us some information as to the angle between the two vectors, There's more… and There's more…. We can use the dot product to find the exact angle between these two vectors. The key to this is an alternate definition of the dot product.

Geometric definition

Given the vectors Geometric definition and Geometric definition, the geometric definition of the dot product is the length of Geometric definition multiplied by the length of Geometric definition multiplied by the cosine of the angle between them:

Geometric definition

The || operator in the above equation means length and will be covered in the next section. We will cover the geometric definition and other properties of the dot product later in this chapter.

End of Section 5
Next Section
bookmark search playlist download
font-size

Change the font size

margin-width

Change margin width

day-mode

Change background colour

Close icon Search
Country selected
Close icon Your notes and bookmarks

Delete Bookmark

Modal Close icon
Are you sure you want to delete it?
Cancel
Yes, Delete