Computer Aided Geometric Design 17 (2000) 297–307 www.elsevier.com/locate/comaid Filling polygonal holes using C 1 cubic triangular spline patches Charles K. Chui a,1 , Min-Jun Lai b,∗ a Department of Mathematics, Texas A&M University, College Station, TX 77843, USA b Department of Mathematics, University of Georgia, Athens, GA 30602, USA Received November 1998; revised September 1999 Abstract We use the method of energy minimization to fill polygonal holes by C 1 cubic triangular spline patches. We implement the method in MATLAB. Several numerical examples are shown.  2000 Elsevier Science B.V. All rights reserved. Keywords: Minimal energy method; Triangular splines; Polygonal holes 1. Introduction In free-form surface modeling or surface design, we often encounter a polygonal hole when assembling several surface patches together. Usually these given surfaces patches are C 1 bicubic patches. We have to find a mending surface patch to fill the hole such that the modified surface is in C 1 or G 1 globally. In some applications, the mending surface patch to fill the hole may be required to satisfy certain interpolation condition. Several researchers have already tested some ideas, e.g., bicubic patches, subdivision algorithm, and etc. See, e.g. (Hahn, 1989; Gregory and Zhou, 1994; Qu, 1996; Zheng and Ball, 1997). In this paper, we propose to use C 1 cubic triangular spline patches to handle the filling problem. Our method can fill any number of sides of polygonal holes. However, we relax the boundary matching conditions a little bit. More precisely, our filling spline surface only matches the boundary values exactly (for bicubic patches which form a polygonal hole) while approximates the normal derivatives on the boundary of the hole. We use C 1 piecewise cubic triangular spline functions to construct the filling surfaces. Certainly there ∗ Corresponding author. E-mail: mjlai@math.uga.edu. Research supported by the National Science Foundation under Grant # DMS-9870178. 1 Research supported by the National Science Foundation GOALI program under Grant # DMI-9634833 and Boeing Company, St. Louis. 0167-8396/00/$ – see front matter  2000 Elsevier Science B.V. All rights reserved. PII:S0167-8396(00)00005-4