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Minimal energy surfaces using parametric splines. (English) Zbl 0873.65008
Summary: We explore the construction of parametric surfaces which interpolate prescribed 3D scattered data using spaces of parametric splines defined on a 2D triangulation. The method is based on minimizing certain natural energy expressions. Several examples involving filling holes and crowning surfaces are presented, and the role of the triangulation as a parameter is explored. The problem of creating closed surfaces is also addressed. This requires introducing spaces of splines on certain generalized triangulations.
Reviewer: Reviewer (Berlin)

MSC:
65D17 Computer-aided design (modeling of curves and surfaces)
65D07 Numerical computation using splines
68U07 Computer science aspects of computer-aided design
Software:
FITPACK; MA28; NSPIV; TOLMIN; Y12M; YSMP
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References:
[1] Alfeld, P., A trivariate clough-tocher scheme for tetrahedral data, Computer aided geometric design, 1, 169-181, (1984) · Zbl 0566.65003
[2] Alfeld, P., Triangular extrapolation, ()
[3] Alfeld, P.; Piper, B.; Schumaker, L.L., An explicit basis for C1 quartic bivariate splines, SIAM J. numer. anal., 24, 891-911, (1987) · Zbl 0658.65008
[4] Alfeld, P.; Schumaker, L.L., On the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1, Numer. math., 57, 651-661, (1990) · Zbl 0725.41012
[5] Alfeld, P.; Schumaker, L.L.; Whiteley, W., The generic dimension of the space of C1 splines of degree d ≥ 8 on tetrahedral decompositions, SIAM J. numer. anal., 30, 889-920, (1993) · Zbl 0774.41012
[6] Celniker, G.; Gossard, D., Deformable curve and surface finite-elements for free-form shape design, Comput. graph., 25, 257-266, (1991), (ACM SIGGRAPH)
[7] Coons, S., Surfaces for computer aided design, (1964), MIT Cambridge, MA, MAC-TR-4
[8] de Boor, C., B-form basics, (), 131-148, Philadelphia, PA
[9] DeRose, T., Necessary and sufficient conditions for tangent plane continuity of Bézier surfaces, Computer aided geometric design, 7, 165-179, (1990) · Zbl 0807.65008
[10] DeRose, T.; Mann, S., An approximately G1 cubic surface interpolant, (), 185-196
[11] Dierckx, P., Curve and surface Fitting with splines, (1993), Clarendon Press Oxford · Zbl 0782.41016
[12] Doo, D.; Sabin, M.A., Behaviour of recursive division surfaces near extraordinary points, Computer-aided design, 10, 350-355, (1978)
[13] Du, W., Etude sur la représentation de surfaces complexes: application à la reconstruction de surfaces échantillonnées, (1988), Télécom Paris
[14] Duff, I.S., MA28—a set of Fortran subroutines for sparse unsymmetric linear equations, ()
[15] Dyn, N.; Ferguson, W., The numerical solution of equality constrained quadratic programming problems, Math. comput., 41, 165-170, (1983) · Zbl 0527.49030
[16] Eisenstat, S.C.; Gursky, M.C.; Schultz, M.H.; Sherman, A.H., Yale sparse matrix package. 1: the symmetric codes, Internat. J. numer. meth. engr., 18, 1145-1151, (1982) · Zbl 0492.65012
[17] Farin, G., Triangular Bernstein-Bézier patches, Computer aided geometric design, 3, 83-127, (1986)
[18] Gmelig Meyling, R.H.J., Approximation by cubic C1 splines on arbitrary triangulations, Numer. math., 51, 65-85, (1987) · Zbl 0595.41010
[19] Goodman, T.N.T., Closed surfaces defined from biquadratic splines, Construct. approx., 7, 149-160, (1991) · Zbl 0732.41010
[20] Gordon, W., Free-form surface interpolation through curve networks, (1969), General Motors Research Labs Warren, MI, GMR-921 · Zbl 0192.42201
[21] Gregory, J.A., N-sided surface patches, (), 217-232
[22] Gregory, J.A.; Hahn, J.M., A C2 polygonal surface patch, Computer aided geometric design, 6, 69-75, (1989) · Zbl 0664.65016
[23] Gregory, J.; Lau, V.K.H.; Zhou, J., Smooth parametric surfaces and n-sided patches, (), 457-498
[24] Hagen, H.; Schulze, G., Automatic smoothing with geometric surface patches, Computer aided geometric design, 4, 231-235, (1987) · Zbl 0635.65011
[25] Hahn, J.M., Geometric continuous patch complexes, Computer aided geometric design, 6, 55-67, (1989) · Zbl 0664.65015
[26] Hahn, J.M., Filling polygonal holes with rectangular patches, (), 81-91
[27] Herron, G., Smooth closed surfaces with discrete triangular interpolants, Computer aided geometric design, 2, 297-306, (1985) · Zbl 0585.65004
[28] Höllig, K.; Mögerle, H., G-splines, Computer aided geometric design, 7, 197-207, (1990) · Zbl 0719.65010
[29] Hosaka, M., Theory of curve and surface synthesis and their smooth Fitting, Information processing in Japan, 9, 60-68, (1969) · Zbl 0276.68042
[30] Hosaka, M.; Kimura, F., Non-four-sided patch expressions with control points, Computer aided geometric design, 1, 75-86, (1984) · Zbl 0581.65009
[31] Kallay, M., Constrained optimization in surface design, (1992), EDS Bellevue, WA
[32] Loop, C.; DeRose, T., A multisided generalization of Bézier surfaces, Trans. graph., 8, 204-234, (1989) · Zbl 0746.68097
[33] Loop, C.; DeRose, T., Generalized B-spline surfaces of arbitrary topology, Siggraph’90, 24, 347-356, (1990)
[34] Lounsbery, M.; Mann, S.; DeRose, T., An overview of parametric surface interpolation, Comput. graph. appl., 12, 5, 45-52, (1992)
[35] Mann, S.; Loop, C.; Lounsbery, M.; Meyers, D.; Painter, J.; DeRose, T.; Sloan, K., A survey of parametric scattered data Fitting using triangular interpolants, (), 145-172 · Zbl 0823.41020
[36] Moreton, H.; Séquin, C., Functional optimization for fair surface design, Comput. graph., 26, 167-176, (1992), (ACM SIGGRAPH)
[37] Neamtu, M., Multivariate splines, () · Zbl 0757.41019
[38] Peters, J., Smooth mesh interpolation with cubic patches, Computer-aided design, 22, 109-120, (1990) · Zbl 0698.65005
[39] Peters, J., Local smooth surface interpolation: a classification, Computer aided geometric design, 7, 191-195, (1990) · Zbl 0713.65003
[40] Peters, J., Smooth interpolation of a mesh of curves, Construct. approx., 7, 221-246, (1991) · Zbl 0726.41011
[41] Peters, J., Parametrizing singularly to enclose data points by a smooth parametric surface, ()
[42] Peters, J., Joining smooth patches around a vertex to form a Ck surface, Computer aided geometric design, 9, 387-411, (1992) · Zbl 0760.65015
[43] Peters, J., Smooth free-form surfaces over irregular meshes generalizing quadratic splines, (1993), Purdue University West Lafayette, IN, CSD-TR-92-063
[44] Piper, B., Visually smooth interpolation with triangular Bézier patches, (), 221-234, Philadelphia, PA
[45] Powell, M.J.D., TOLMIN: A Fortran package for linearly constrained optimization calculations, (1989), Dept. of Applied Mathematics and Theoretical Physics, University of Cambridge · Zbl 0695.90084
[46] Quak, E.; Schumaker, L.L., Calculation of the energy of a piecewise polynomial surface, (), 134-143 · Zbl 0746.41012
[47] Reif, U., Neue aspekte in der theorie der freiformflächen beliebiger topologie, () · Zbl 0850.65021
[48] Sabin, M., Non-rectangular surface patches suitable for inclusion in a B-spline surface, (), 57-69
[49] Sarraga, R., G1 interpolation of generally unrestricted cubic Bézier curves, Computer aided geometric design, 4, 23-39, (1987) · Zbl 0621.65002
[50] Sarraga, R., Errata: G1 interpolation of generally unrestricted cubic Bézier curves, Computer aided geometric design, 6, 167-172, (1989) · Zbl 0669.65004
[51] Sarraga, R., Computer modelling of surfaces with arbitrary shapes, Comput. graph. appl., 10, 2, 67-77, (1990)
[52] Schmidt, R., Eine methode zur konstruktion von \(C\^{}\{1\}- Flächen\) zur interpolation unregelmässig verteilter daten, (), 343-361
[53] Schumaker, L.L., On the dimension of spaces of piecewise polynomials in two variables, (), 396-412
[54] Schumaker, L.L., Recent progress on multivariate splines, (), 535-562
[55] Seager, M.K., Sparse linear algebra package version 2.0, (1989), Lawrence Livermore National Laboratory
[56] Sherman, A.H., Algorithm 533. NSPIV, a Fortran subroutine for sparse Gaussian elimination with partial pivoting, ACM trans. math. software, 4, 391-398, (1978)
[57] Shirman, L.; Séquin, C., Local surface interpolation with Bézier patches, Computer aided geometric design, 4, 279-295, (1987) · Zbl 0638.65006
[58] Shirman, L.; Séquin, C., Local surface interpolation with Bézier patches: errata and improvements, Computer aided geometric design, 8, 217-221, (1991) · Zbl 0745.65013
[59] Storrey, D.J.T.; Ball, A.A., Design of an n-sided patch from Hermite boundary data, Computer aided geometric design, 6, 111-120, (1989) · Zbl 0671.65010
[60] Timoshenko, S.; Woinowsky-Krieger, S., Theory of plates and shells, (1959), McGraw-Hill New York · Zbl 0114.40801
[61] Van Wijk, J.J., Bicubic patches for approximating non-rectangular control-point meshes, Computer-aided design, 3, 1-13, (1986) · Zbl 0616.65010
[62] Varady, T., Survey and new results in n-sided patch generation, (), 203-235
[63] von Golitschek, M.; Schumaker, L.L., Data Fitting by penalized least squares, (), 210-227 · Zbl 0749.41004
[64] Zlatev, Z.; Wasniewski, J.; Schaumburg, K., Y12M. solution of large and sparse systems of linear algebraic equations, () · Zbl 0461.65023
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