Non-Distorted Texture Mapping Using Angle Based Flattening

This article introduces a new method for surface parameterization for texture mapping. In the first step of the algorithm we compute a planar triangulation of the faceted surface, using an Angle Based Flattening (ABF) formulation. This provides a mapping function from a three-dimensional surface to the plane. The ABF formulation defines the flattening problem as a constrained minimization problem in terms of angles (only). After generating the flat triangulation we compute a mapping from the flat triangulation to a rectangular region. The mapping minimizes the distortion of distances and areas in the parameterization with respect to the original three-dimensional surface. The texture mapping function is generated by combining the two mapping functions. The method is more general and robust than previous methods and is guaranteed to generate a valid solution. It can handle any surface for which a connected, valid, two-dimensional parameterization exists, including surfaces with large curvature gradients. It does not require the boundary of the flat two-dimensional domain to be predefined or convex. The flattening step of the algorithm uses the necessary and sufficient constraints for a valid two-dimensional triangulation. As a result, the existence of a theoretical solution to the flattening procedure is guaranteed. The second step of the algorithm aims at the preservation of distances in the combined mapping, resulting in a minimal distortion texture mapping.