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!
Game Physics Cookbook
Credits
Acknowledgements
Acknowledgements
www.PacktPub.com
Customer Feedback
Preface
Free Chapter
Vectors
Matrices
Matrix Transformations
2D Primitive Shapes
2D Collisions
2D Optimizations
3D Primitive Shapes
3D Point Tests
3D Shape Intersections
3D Line Intersections
Triangles and Meshes
Models and Scenes
Camera and Frustum
Constraint Solving
Manifolds and Impulses
Springs and Joints
Index

Projection

Sometimes it's useful to decompose a vector into parallel and perpendicular components with respect to another vector. Projecting onto will give us the length of in the direction of . This projection decomposes into its parallel component with respect to . Once we know the parallel component of , we can use it to get the perpendicular component. The formula for projecting onto is as follows:

The perpendicular component of with respect to is defined as follows:

Implementing the projection is fairly straightforward as we already have both the dot product and magnitude squared defined. In the following function, the vector being projected is represented by the variable `length`, and the vector it is being projected onto is represented by the variable `direction`. If we compare it to the preceding formula, `length` is , and `direction` is .

How to do it…

Follow these steps to implement projection functions for two and three dimensional vectors. A function to get the perpendicular component of the projection is also described:

1. Declare the projection and perpendicular functions in `vectors.h`:

```vec2 Project(const vec2& length, const vec2& direction);
vec3 Project(const vec3& length, const vec3& direction);

vec2 Perpendicular(const vec2& len, const vec2& dir);
vec3 Perpendicular(const vec3& len, const vec3& dir);```
2. Add the implementation of projection to `vectors.cpp`:

```vec2 Project(const vec2& length, const vec2& direction) {
float dot = Dot(length, direction);
float magSq = MagnitudeSq(direction);
return direction * (dot / magSq);
}

vec3 Project(const vec3& length, const vec3& direction) {
float dot = Dot(length, direction);
float magSq = MagnitudeSq(direction);
return direction * (dot / magSq);
}```
3. Add the implementation of perpendicular to `vectors.cpp`:

```vec2 Perpendicular(const vec2& len, const vec2& dir) {
return len - Project(len, dir);
}

vec3 Perpendicular(const vec3& len, const vec3& dir) {
return len - Project(len, dir);
}```

How it works…

Let's explore how projection works. Say we want to project onto , to find . Having a ' character next to a vector means prime; it's a transformed version of the vector; is pronounced A-Prime:

From the preceding figure we see that can be found by subtracting some unknown vector from . This unknown vector is the perpendicular component of with respect to , let's call it :

We can get the perpendicular component by subtracting the projection of onto from. The projection at this point is still unknown, that's what we are trying to find:

Because points in the same direction as , we can express as scaling by some unknown scalar s, . Knowing this, the problem becomes, how do we find s?:

The dot product of two perpendicular vectors is 0. Because of this, the dot product of and is going to be 0:

Substitute the value of with the equation we use to find its value, :

Finally, let's substitute with the equation we use to find its value, :

Now the only unknown in the formula is s, let's try to find it. The dot product exhibits the distributive property, let's distribute :

Let's start to isolate s, first we add to both sides of the equation:

Now we can isolate s if we divide both sides of the equation by . Remember, the dot product of a vector with itself yields the square magnitude of that vector:

Now we can solve by substituting s with the preceding formula. The final equation becomes: