#### 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 (27 chapters)
Game Physics Cookbook
Credits
About the Author
Acknowledgements
About the Reviewer
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
Advanced Topics
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:

### 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:

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 and :

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

### 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 and , 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