How to orient a surface normal to a cubic surface

Question

The immediate question is: in HLSL, how can I orient a surface normal generated in UV space so that I can apply it to a cube face?

The overall project is that I'm trying to build a procedural planet generator. I'm using a cube projected to a sphere where each face is a quad-tree. The intermediate project is that I want to generate a normal map for the terrain. I think the best way for me to do this (and understand what I'm doing) is to generate the normals for the cube version first, then work out the transformations to warp those normals to the sphere.

Ideally the method would not involve branching code based on which face I'm working on (i.e., if we're on the top side of the cube, then U = X and V = Y). In other words, what I'm hoping for is some math-magic like "Oh, just cross multiply the terrain normal by the dot product of the cube face normal and blah blah blah".

9/20/10 ETA:

I know how to calculate normals for a flat surface. My subsequent problem is two-fold:

1. How do I rotate the normal map so it is oriented correctly on each of the cube faces?
2. how do I warp the flat normal map so that it wraps the sphere?

I've found one solution that uses a Jacobian matrix, but I can't get it to work. Even when all of the normals are pointing straight up (i.e. a flat surface), the HLSL code involving the Jacobian totally messes up the lighting--so it makes me not trust my implementation of the solution.

Answer 1

Your normals should be in tangent space, that way you can apply them onto any surface for which you can create a tangent space (trivial for spheres.) Basically the normal map describes how the normal differs from the actual normal on a small patch on the surface. The tangent space is what you probably try to get with your Jacobian matrix -- for a sphere, you can create a tangent space coordinate system at any given point by using the normal at the point and two perpendicular vectors -- just orient them consistently (for instance, along u and v) and you're done. You can then translate your incoming light vector into tangent space (or the other way round) and light with the new normal. The advantage is that your normal map will work on any object with a defined tangent space and UV mapping.

On the cube, your tangent space for each face is simply that face itself (i.e. if you have a face which has a normal -Z for instance, and your tangent space up is +Y, you just rotate that normal using a matrix which maps -Z to +Y.)

Answer 2

"I know how to calculate normals for a flat surface." Exactly how are you doing this? As far as I know, you shouldn't have to do anything to map it to a sphere other than map your UV's to the sphere surface correctly. Normal maps are always in a "flat" plane because the texture itself is "flat". The shape of the object that you apply the normal map to shouldn't matter.