Normals

Normals are vectors that are perpendicular to the surface we want to illuminate. Normals represent the orientation of the surface and are therefore critical to modeling the interaction between a light source and the object. Given that each vertex has an associated normal vector, we can use the cross product to calculate normals.

Let's break this down. If we have the triangle conformed by vertices p0, p1, and p2, we can define the v1 vector as p1 - p0 and the v2 vector as p2 - p0. The normal is then obtained by calculating the v1 x v2 cross-product. Graphically, this procedure looks something like the following:

We then repeat the same calculation for each vertex on each triangle. But, what about the vertices that are shared by more than one triangle? Each shared vertex normal will receive a contribution from each of the triangles in which the vertex appears.

For example, say that the p1 vertex is shared by the #1 and #2 triangles, and that we have already calculated the normals for the vertices of the #1 triangle. Then, we need to update the p1 normal by adding up the calculated normal for p1 on the #2 triangle. This is a vector sum. Graphically, this looks similar to the following:

Similar to lights, normals are generally normalized to facilitate mathematical operations.