Filling N-sided Holes

Smooth surface patches, such as Gregory patch, Brown’s square and Nielson- Foley patch, which interpolate a given function and its derivatives on the boundary of a rectangle or a triangle, with incompatible twist terms, have been constructed with rational parametric representation by using boolean sum techniques, convex combination methods and procedural methods to fill N-sided holes. Chiyokura and Kimura proposed a representation of Gregory patch in Bernstein-Bezier form, where interior control points are expressed as convex combinations of incompatible control points via rational blending functions. No such representation is known for solutions to the above problem over pentagonal domains. We construct smooth rational surface patches which interpolate a given function and its cross-boundary Ck derivatives on the boundary of any convex polygonal domain with incompatible twist data. These patches are represented in S-patch form, where control points are expressed as convex combinations of incompatible control points via rational blending functions. This constructive rational technique provides novel solutions for blending incompatible Ck data over polygonal domains. In particular, new solutions are constructed for rectangular and triangular patches as well.