A Multi-sided Bézier Patch with a Simple Control Structure

• Published in: Computer graphics forum Vol. 35; no. 2; pp. 307 - 317
• Main Authors: Várady, Tamás, Salvi, Péter, Karikó, György
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.05.2016
• Subjects: Algorithms; Analysis; and object representations; Bezier; Bezier patches; Boundaries; Categories and Subject Descriptors (according to ACM CCS); Displacement; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Curve; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations; Image processing systems; Mathematical analysis; Mathematical models; Representations; solid; Studies; surface; Topological manifolds
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/cgf.12833

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.