Multi-sided Bézier surfaces over curved, multi-connected domains

Abstract

A new multi-sided, control point based surface formulation is proposed, extending the principles of the Generalized Bézier patch published by Várady et al. (2016). The surface is constructed from side interpolants given in Bézier form. The boundary curves may have different degrees, and the cross-derivatives can ensure arbitrary G-continuity with adjacent patches. The representation is capable of handling boundary curves with high curvature variations and concave angles. The surfaces are C∞-continuous and may have holes in their interior.The most distinctive feature of the patch is that it is defined over a planar domain with curved boundaries that mimic the shape of the 3D boundary curves. Local parameters, derived from harmonic barycentric coordinates, are associated with each side of the domain. The control points are multiplied by Bernstein functions and additional rational terms that provide the same degree of freedom for shape design as their quadrilateral counterparts.The paper introduces the full construction of Curved Domain (CD) Bézier patches including domain generation, parameterization, basis functions, and methods for additional interior shape control. Specific problems related to CD surfaces are also discussed with many examples that demonstrate the peculiarity and usefulness of the scheme.