A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations. (English) Zbl 1181.65022

Summary: We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to \(C^{1}\) cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods.


65D17 Computer-aided design (modeling of curves and surfaces)
41A15 Spline approximation
41A63 Multidimensional problems
65D07 Numerical computation using splines
65D15 Algorithms for approximation of functions


GMT; pchip; TSFIT
Full Text: DOI


[1] Ambrosetti, A.; Prodi, G., A primer of nonlinear analysis, (1993), Cambridge University Press Cambridge, UK · Zbl 0781.47046
[2] Carlson, R.E.; Fritsch, F.N., Monotone piecewise bicubic interpolation, SIAM J. numer. anal., 22, 386-400, (1985) · Zbl 0571.65005
[3] Costantini, P., Curve and surface construction using variable degree polynomial splines, Comput. aided geom. design, 24, 426-446, (2000) · Zbl 0938.68128
[4] Costantini, P.; Manni, C., A parametric cubic element with tension properties, SIAM J. numer. anal., 36, 607-628, (1999) · Zbl 0924.65008
[5] Costantini, P.; Manni, C., Comonotone parametric \(C^1\) interpolation of non-gridded data, J. comput. appl. math., 75, 147-169, (1996) · Zbl 0864.65005
[6] Davydov, O.; Morandi, R.; Sestini, A., Local hybrid approximation for scattered data Fitting with bivariate splines, Comput. aided geom. design, 23, 703-721, (2006) · Zbl 1171.65317
[7] Davydov, O.; Sestini, A.; Morandi, R., Local RBF approximation for scattered data Fitting with bivariate splines, (), 91-102 · Zbl 1081.65502
[8] Davydov, O.; Zeilfelder, F., Scattered data Fitting by direct extension of local polynomials to bivariate splines, Adv. comput. math., 21, 223-271, (2004) · Zbl 1065.41017
[9] O. Davydov, F. Zeilfelder, TSFIT: A Software Package for Two-Stage Scattered Data Fitting, 2005 available under GPL from http://www.maths.strath.ac.uk/ aas04108/tsfit/
[10] Delbourgo, R., Gregory shape preserving piecewise rational interpolation, SIAM J. sci. stat. comput., 6, 967-976, (1985) · Zbl 0586.65006
[11] Edelman, A.; Micchelli, C.A., Admissible slopes for monotone and convex interpolation, Numer. math., 51, 441-458, (1987) · Zbl 0624.41001
[12] Fraeijs de veubeke, B., A conforming finite element for plate bending, J. solids struct., 4, 95-108, (1968) · Zbl 0168.22602
[13] Fritsch, F.N.; Carlson, R.E., Monotone piecewise cubic interpolation, SIAM J. numer. anal., 17, 238-246, (1980) · Zbl 0423.65011
[14] T.N.T. Goodman, Shape preserving interpolation by curves, in: J. Levesley, I. J.Anderson and J. C. Mason, (Eds.), Algorithms for Approximation IV, 2002, pp. 24-35, Huddersfield: University of Huddersfield Proceedings
[15] Haber, J.; Zeilfelder, F.; Davydov, O.; Seidel, H.-P., Smooth approximation and rendering of large scattered data sets, IEEE visualization, 571, 341-347, (2001)
[16] Han, L.; Schumaker, L.L., Fitting monotone surfaces to scattered data using \(C^1\) piecewise cubics, SIAM J. numer. anal., 34, 569-585, (1997) · Zbl 0881.41010
[17] Hoschek, J.; Lasser, D., Fundamentals of computer aided geometric design, (1993), A. K. Peters Wellesley, Massachusetts · Zbl 0788.68002
[18] Koch, P.E.; Lyche, T., (), 173-190
[19] Kvasov, B.I., Algorithms for shape preserving local approximation with automatic selection of tension parameters, Comput. aided geom. design, 17, 17-37, (2000) · Zbl 0939.68122
[20] Lai, M.-J.; Schumaker, L.L., Spline functions on triangulations, (2007), Cambridge University Press Cambridge · Zbl 1185.41001
[21] Manni, C., \(C^1\) comonotone Hermite interpolation via parametric cubics, J. comput. appl. math., 69, 143-157, (1996) · Zbl 0858.65007
[22] Manni, C., A general parametric framework for functional tension schemes, J. comput. appl. math., 119, 275-300, (2000) · Zbl 0963.65016
[23] Sander, G., Bornes supérieures et inférieures dans l’analyse matricielle des plaques en flexion-torsion, J. bull. soc. royale sci. liège, 33, 456-494, (1964)
[24] Schweikert, D.G., An interpolation curve using a spline in tension, J. math. phys., 45, 312-317, (1966) · Zbl 0146.14102
[25] Smith, W.H.F.; Wessel, P., Gridding with continuous curvature splines in tension, Geophysics, 55, 293-305, (1990)
[26] W.H.F. Smith, P. Wessel, GMT - The Generic Mapping Tools, http://gmt.soest.hawaii.edu/
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.