Jüttler, Bert Surface fitting using convex tensor-product splines. (English) Zbl 0883.65007 J. Comput. Appl. Math. 84, No. 1, 23-44 (1997). The author circumvents the very difficult problem of insuring that a given tensor-product spline surface be convex (or concave) by concentrating on a sufficient condition, insuring that a two-by-two matrix representing the Hessian be definite. He shows that this condition is asymptotically necessary, i.e., in the limit of a continuous surface will describe Dupin’s indicatrix. The paper is highly recommended since it gives all details for implementing the procedure (as a quadratic optimization problem). Examples are given to show that even in cases where the surface has to be divided into convex and concave parts, the procedure of approximation by certainly convex or concave parts by the author’s method leads to an elimination of all kinds of undesirable wiggles in the tensor surface. Reviewer: H.Guggenheimer (West Hempstead) Cited in 8 Documents MSC: 65D17 Computer-aided design (modeling of curves and surfaces) 65D10 Numerical smoothing, curve fitting Keywords:surface fitting; tensor-product spline surface; Dupin’s indicatrix; quadratic optimization Software:FITPACK; lp_solve × Cite Format Result Cite Review PDF Full Text: DOI References: [1] M. Berkelaar, lp-solve, available via anonymous ftp from ftp.es.ele.tue.nl at /pub/lp-solve.; M. Berkelaar, lp-solve, available via anonymous ftp from ftp.es.ele.tue.nl at /pub/lp-solve. [2] J.M. Carnicer, M.S. Floater, J.M. Peña, Linear convexity conditions for triangular and rectangular Bernstein-Bézier surfaces, to appear in Comput. Aided Geom. Design.; J.M. Carnicer, M.S. Floater, J.M. Peña, Linear convexity conditions for triangular and rectangular Bernstein-Bézier surfaces, to appear in Comput. Aided Geom. Design. [3] Carnicer, J. M.; Dahmen, W., Convexity preserving interpolation and Powell-Sabin elements, Comput. Aided Geom. Design, 9, 279-289 (1992) · Zbl 0760.65005 [4] Chang, G.; Feng, Y., An improved condition for the convexity of Bernstein-Bézier polynomials over triangles, Comput. Aided Geom. Design, 1, 279-283 (1984) · Zbl 0563.41009 [5] Chang, G.; Davis, P. J., The convexity of Bernstein polynomials over triangles, J. Approx. Theory, 40, 11-28 (1984) · Zbl 0528.41005 [6] Dahmen, W., Convexity and Bernstein-Bézier polynomials, (Laurent, P. J.; Le Méhauté, A.; Schumaker, L. L., Curves and Surfaces (1991), Academic Press: Academic Press Boston), 107-134 · Zbl 0735.41005 [7] Dierckx, P., An algorithm for cubic spline fitting with convexity constraints, Computing, 24, 349-371 (1980) · Zbl 0419.65006 [8] Dierckx, P., Curve and Surface Fitting with Splines (1993), Clarendon Press: Clarendon Press Oxford · Zbl 0782.41016 [9] Farin, G., Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide (1990), Academic Press: Academic Press Boston · Zbl 0702.68004 [10] Fletcher, R., Practical Methods of Optimization (1990), Wiley-Interscience: Wiley-Interscience Chichester · Zbl 0905.65002 [11] Floater, M., A weak condition for the convexity of tensor-product Bézier and B-spline surfaces, Adv. Comput. Math., 2, 67-80 (1994) · Zbl 0828.65011 [12] Hoschek, J.; Lasser, D., Fundamentals of Computer Aided Geometric Design AK Peters (1993), Wellesley, MA · Zbl 0788.68002 [13] Mulansky, B.; Schmidt, J. W.; Walther, M., Tensor product spline interpolation subject to piecewise bilinear lower and upper bounds, (Hoschek, J.; Kaklis, P., Advanced Course on FAIRSHAPE (1996), Teubner: Teubner Stuttgart), 201-216 · Zbl 0871.41002 [14] Schelske, H. J., Lokale Glättung segmentierter Bézierkurven und Bézierflächen, (Dissertation (1984), TH Darmstadt) [15] Schmidt, J. W., Results and problems in shape preserving interpolation and approximation with polynomial splines, (Schmidt, J. W.; Späth, H., Splines in Numerical Analysis. Splines in Numerical Analysis, Mathematical Research, vol. 52 (1989), Akademie-Verlag: Akademie-Verlag Berlin), 159-170 · Zbl 0677.65009 [16] Schmidt, J. W.; Heß, W., Positivity of cubic polynomials on intervals and positive spline interpolation, BIT, 28, 340-352 (1988) · Zbl 0642.41007 [17] R. Vanderbei, LOQO, available via anonymous ftp from elib.zib-berlin.de at /pub/opt-net/software/loqo/1.08.; R. Vanderbei, LOQO, available via anonymous ftp from elib.zib-berlin.de at /pub/opt-net/software/loqo/1.08. [18] Willemans, K.; Dierckx, P., Surface fitting using convex Powell-Sabin splines, J. Comput. Appl. Math., 56, 263-282 (1994) · Zbl 0827.65017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.