Question

Assume that a 3D polygon is tranformed by a transformation matrix T. Assume the normal vector of the polygon is n. Can we use the formula Tn to compute the normal vector of the transformed polygon? If not, drive the correct formula.

Answer 1

When lighting is enabled in OpenGL, the normal vectors are used to determine how much light is received at the specified vertex or surface. This lighting processing is performed at eye coordinate space, therefore, normal vectors in object coordinates are also transformed to eye coordinates with GL_MODELVIEW matrix.

However, normals vectors are transformed in different way as vertices do. We cannot simply multiply GL_MODELVIEW matrix and normal. Consider a normal vector (1,0,0) at vertex (0,0,0). If GL_MODELVIEW matrix is simply translating 2 unit up along Y-axis, then the vertex coordinates will be (0,2,0). But, the normal should be remain same as (1,0,0), not (1,2,0).

In order to understand how normal vectors are transformed to eye space, think the normals as coefficients of plane equations, which are perpendicular to the planes.

OpenGL Surface Normal Vector

Imagine a triangle polygon with 3 vertices; v₁-v₂-v₃, and the normal of this surface is as a homogeneous plane.(For Euclidean space, the normal is .) If we think of the triangle as a homogeneous plane, then the plane equation becomes;

Since 3 vertices lie on this plane, the plane equation is also true when we substitute these vertices to the equation. For example, for , it satisfies;

The equivalent matrix form of the plane equation is;

The plane equation is multiplying the transposed normal (n^T) and the vertex together.
Now, we modify the above equation to acquire normal vector transformation formula by inserting GL_MODELVIEW matrix M^-1M in between;
(The following equation is still equivalent to the above because M^-1M is identity matrix.)

As you can see, the right part of the above equation is transforming vertex to eye space, and the left part is going to be the normal vector in eye space because the plane equation is also transformed. It reads as"The transformed vertex lies on the transformed plane in eye coordinates space".

Therefore, transforming normal from object space to eye space with GL_MODELVIEW matrix M is;

Or, by converting pre-multiplication to post-multiplication form, we get;