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A Multi‐sided Bézier Patch with a Simple Control Structure
Author(s) -
Várady Tamás,
Salvi Péter,
Karikó György
Publication year - 2016
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/cgf.12833
Subject(s) - quadrilateral , bézier curve , mathematics , surface (topology) , degree (music) , simple (philosophy) , base (topology) , topology (electrical circuits) , representation (politics) , geometry , computer science , algorithm , mathematical analysis , combinatorics , philosophy , physics , epistemology , finite element method , politics , political science , acoustics , law , thermodynamics
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 G 1 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.