Triangular tile mapping on freeform surface

I have a known freeform curvature that is in a step or iges file. I would then like to map a triangle (equilateral, right angle) across the surface. The difficult part is the triangle edge length and angles in a 3D space must remain the same. Picture I want to plaster a triangular tile on a curve surface. I want to determine the best size and shape to conform to the surface as close as possible.

[Sean de Wolski]

Can you provide sample data? It sounds like a standard optimization problem.