Triangular Berstein-bezier Patches, a Survey and New Results

Triangular Bernstein-Bezier patches are an alternative to the standard rectangular ones. They are maps of triangular domains into R cubed, where the patches are described by control nets similar to the rectangular case. One difference in the definition of surface normals at the patch corners: rectangular patches have inherent problems with so-called degenerate corners, i.e., edges collapsing into one point. Triangular patches do not have these problems. Since triangular shapes arise rather frequently in the design of complicated surfaces (e.g., interior car body panels), it seems that for such surfaces triangular patches are a promising addition to the usually employed rectangular patches. Another major application is the formulation of interpolants to solve the scattered data problem: here Bernstein-Bezier patches provide a very powerful tool; they can be used to discover new schemes as well as to discover new properties of existing methods.