Calculating the Normal Matrix in OpenGL

Question

The following site says to use the model_view matrix when computing the normal matrix (assuming we are not using the built in gl_NormalMatrix): (site)Light House. I have the following algorithm in my program:

//Calculate Normal matrix

// 1. Multiply the model matrix by the view matrix and then grab the upper left
// corner 3 x 3 matrix.
mat3x3 mv_orientation = glext::orientation_matrix<float, glext::column>(
  glext::model_view<float, glext::column>(glext_model_, glext_view_));

// 2. Because openGL matrices use homogeneous coordinate an affine inversion
// should work???
mv_orientation.affine_invert();

// 3. The normal matrix is defined as the transpose of the inverse of the upper
// left 3 X 3 matrix
mv_orientation.transpose();

// 4. Place this into the shader
basic_shader_.set_uniform_3d_matrix("normal_matrix", mv_orientation.to_gl_matrix());

Assuming most statements above are correct in the aforementioned code. Do you not include the projection matrix in the computation of the normal matrix? If not why, does the projection matrix not affect the normals like they do points?

Answer 1

That's because projection is not an affine transformation. Projections don't maintain the inner product and then they don't maintain the angles. And the real angles that have effect on the light diffusion and reflection are the angles in the affine 3d space. So using also the projection matrix would get you different angles, wrong angles, and hence wrong lights.

Answer 2

Do you not include the projection matrix in the computation of the normal matrix?

No. Normals are required for calculations, like illumination, happening in world and/or view space. It doesn't make sense from a mathematical point of you to do this after projection.

If not why, does the projection matrix not affect the normals like they do points?

Because it would make no sense. That normals should not undergo projective transformation was the original reason to have a separate projection matrix. If you'd put normals through the projection they'd loose their meaning and usefullness.