Filling N-sided Regions With Bicubically Blended Coons Patches

Filling N-sided regions is an essential operation in shape and surface modeling,which iswidely used in vertex blending, filling surface generation and other modeling phases. It aimsto provide a trimmed surface or surface patches in polynomial or rational B-spline form,which interpolate the hole’s boundary and cross-boundary derivative curves, fulfillinggeometric continuity conditions on the boundary. Positional and tangential continuities arehighly required in designing and manufacturing.Since B-spline has become a popular andstandard representation for3D models in CAD/CAM systems, filling N-sided holes usingB-spline surfaces is widely discussed in the past two decades.This paper presents an algorithm to fill an n-sided region with NURBS patches thatinterpolate boundary curves and approximate given cross-boundary derivatives. Thealgorithm first computes a central point, a central normal, the central, and the cornerderivative vectors. Then the region is split into n quadrilateral patches by connecting a centralvertex of the region to the middle point of each boundary curve.The NURBS surfaces joiningalong inner or boundary curves have normal vectors that do not deviate more than theuser-specified angular tolerance ε. The method is general in that there are no restrictions onthe number of boundary curves, and the cross-boundary derivatives can be specifiedindependently. Also, the variational approach to solving such problems has receivedconsiderable attention in the past few years, due to their efficiency and usefulness in thefitting and design of curves and surfaces.