A Multi-sided Bézier Patch with a Simple Control Structure

Abstract

A new n-sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding crossderivatives are specified as conventional Bézier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented. This Generalized Bézier (GB) patch is based on a new displacement scheme that builds up multi-sided patches as a combination of a base patch, n displacement patches and an interior patch; this is considered to be an alternative to the Boolean sum concept. The input ribbons may have different degrees, but the final patch representation has a uniform degree. Interior control points- other than those specified by the user-are placed automatically by a special degree elevation algorithm. GB patches connect to adjacent Bézier surfaces with G1 continuity. The control structure is simple and intuitive; the number of control points is proportional to those of quadrilateral control grids. The scheme is introduced through simple examples; suggestions for future work are also discussed.