Planar mapping/projecting UV coordinates from normal

Question

I've got set of vertices, and would like to calculate their UV coordinates by projecting them by a given normal. Similar to what "planar mapping" does in 3ds max.

Another way of looking at it is that I want to flatten the vertex positions from a given normal direction, so that I can simply use their X & Y (or X & Z depending on your preference) coordinates as UV coordinates.

I'm thinking it's a simple matter of a mathematical Vector function, basically rotating vertex positions along the normal, but I can't figure out the math behind it. I'm using C# and XNA, so I've got Vector3, Matrix and Quaternions. Preferably as cheap as possible.

Any help?

Answer 1

First you construct two basis vectors perpendicular to your projection axis P.

Because of the Hairy Ball Theorem, there's no standard & completely continuous way to do this, but we can hack around it something like...

U = cross(P, (0, 0, 1));
if (dot(U, U) < 0.001)
    U = (1, 0, 0);
else
    normalize(U);

V = normalize(cross(P, U));

Then for each point Q in your mesh, you can construct a u,v coordinate by projecting it onto each basis vector:

uv = (dot(Q, U), dot(Q, V)) * uvscale;

With a scalar multiply if you want your uvs larger/smaller.

This is equivalent to a matrix multiplication, where U and V are the first two rows of a rotation matrix, and the third row P is omitted.

I may have gotten the handedness wrong above, so if your uvs come out mirrored from what you expect, flip the arguments to cross()