Convert UV coord to 3d position

Okay, I think I know how to do this.

Roughly, first treat the UVs of the vertices as a 2d space, and find the barycentric coords of the input UV. This gives an output UV s.t.
v0.uv + v1.uv*outU + v2.uv*outV = inUV

then outUV can be applied to the vertices positions like:

outPos = v0.xyz + v1.xyz*outU + v2.xyz*outV

and that's that. So just need the 2d triangle barycentric calculation.

You're on the right track. What you would do is find the barycentric coordinates of the input point in world space (or whatever other space the triangle is in), and then apply those coordinates to texture space.

The trick is to translate the triangle so that one of the vertices (call it v0) is at the origin. We're allowed to do this because the barycentric coordinates are translationally invariant. This gives you the vectors (v1 - v0) and (v2 - v0) for two edges of the triangle (enough to define a planar subspace), and the input point (p - v0). Now let's plug (v1 - v0) and (v2 - v0) into the columns of a 3x2 matrix called 'A', and rename (p - v0) to 'b'. We can now use the handy projection formula (A^TA)^-1A^Tb, which will return a 2x1 vector giving us the barycentric coordinates for v1 and v2. The coordinate for v0 is just 1 - v1 - v2. Note that since 'A' is a tiny matrix, you can expand the math for the matrix multiplication and inverse fairly easily into a bunch of dot products.

You can then just use those same barycentric coordinates in UV space to determine the texture coordinate.

if your texture coordinates (u,v) are unique over the surface of your object, you can do something like this. Render your 3d object, and in the vertex shader output the texture coordinates as the position (they will be between 0 to 1, put them in clip-space: -1 to 1). Output the actual 3D position of the vertex as texture coordinates so that they get interpolated across the fragments. And thats it. you will have all the in-between 3D positions.

Of course, this is assuming that you are trying to do something like wrapping a texture around your object that stores 3D positions at each pixel, which is what most applications are trying to do (or something similar). However if your purpose is something else, I guess you'll have to do the interpolation manually.

It sounds like something to do with lightmapping or normal mapping.