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

The reflected vector can be found with the following formula:

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

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 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 and , we're going to find , which is the reflection of around :

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

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

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

This is how we get to the final formula, .