A new n-sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding cross-derivatives 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 G1continuity. 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.
Citation
Várady, T., Salvi, P., & Karikó, G. (2016). A Multi-sided Bézier Patch with a Simple Control Structure. Computer Graphics Forum, 35(2), 307–317. doi:10.1111/cgf.12833
Acknowledgements
This is a joint work by researchers at the Budapest University of Technology and Economics and a small technology company ShapEx Ltd., Budapest. The pictures were generated by a prototype system called Sketches. The dolphin model has been provided for us by Cindy Grimm (Washington University); the face model was designed by Supriya Chewle (KAUST, Saudi Arabia). The project was partially supported by the Hungarian Scientific Research Fund (OTKA, No. 101845).