Multi-sided Bézier surfaces over concave polygonal domains

Abstract

A new multi-sided, control point based surface representation is introduced, based on the Generalized Bézier patch [1]. While the original surface is based on convex polygonal domains and a specific, uniform arrangement of control points, the new construction permits domains with concave angles and supports a more general control point structure, where independent “half-Bézier” interpolants, or ribbons, are blended together. The ribbons may have arbitrary degrees along the boundary and also in the cross-derivative direction; the related control points ensure tangent- or curvature-continuous connection to adjacent quadrilateral Bézier patches and permit shape editing and optimization, when needed.The surface comprises four components: (i) a concave domain generated from a 3D loop of boundary edges, (ii) half-Bézier interpolants, (iii) parameterizations that cover the full domain for each interpolant, and (iv) blending functions that guarantee both Bézier-like behavior along the boundaries and a smooth, C∞-continuous composition in the interior of the patch. Editing concave Bézier patches using additional control points is also discussed. A few interesting test examples illustrate the benefits of the method.