Abstract
The application of Powell-Sabin’s or Clough-Tocher’s schemes to scattered data problems, as known requires the knowledge of the partial derivatives of first order at the vertices of an underlying triangulation. We study alocal method for generating partial derivatives based on the minimization of the energy functional on the star of triangles sharing a node that we called acell. The functional is associated to some piecewise polynomial function interpolating the points. The proposed method combines theglobal Method II by Renka and Cline (cf. [16, pp. 230–231]) with the variational approach suggested by Alfeld (cf. [2]) with care to efficiency in the computations. The locality together with some implementation strategies produces a method well suited for the treatment of a big amount of data. An improvement of the estimates is also proposed.
Similar content being viewed by others
References
Akima, H.: On estimating partial derivatives for bivariate interpolation of scattered data. Rocky Mountain J. Math.14, 41–52 (1984).
Alfeld, P.: Derivative generation from multivariate scattered data by functional minimization. Comput. Aided Geom. Design2, 281–296 (1985).
Barnhill, R. E., Farin, G.:C 1 quintic interpolation over triangles: Two explicit representations. Int. J. Numer. Methods Eng.17, 1763–1778 (1981).
Cline, A. K., Renka, R. J.: A constrained two-dimensional triangulation and the solution of closest node problems in the presence of barriers. SIAM J. Numer. Anal.27, 1305–1321 (1990).
Clough, R. W., Tocher, J. L.: Finite element stiffness matrices for analysis of plates in bending. In: Conference on Matrix Methods in Structural Mechanics, Ohio, 1965. Wright Patterson A. F. B.
De Marchi, S., Fasoli, D., Morandi Cecchi, M.: On representing the Venice’s lagoon bed by aC 1 interpolating surface. Submitted to BIT.
De Marchi, S., Fasoli, D., Morandi Cecchi, M.: LABSUP: a LAboraty for BivariateC 1 SUrfaces and Patches. Technical Report, 10/96. University of Padova, Italy, 1996.
Farin, G.: Triangular Bernstein-Bézier patches. Comput. Aided Geom. Design3, 83–127 (1986).
Farin, G.: Curves and surfaces for CAGD. A practical guide, 3rd ed. London: Academic Press, 1993.
Fasoli, D.: InterpolazioneC 1 di dati scattered e procedure per la stima delle derivate parziali. Master’s thesis, University of Padova, 1995 (in Italian).
Goodman, T. N. T., Said, H. B., Chang, L. H. T.: Local derivative estimation for scattered data interpolation. Appl. Math. Comp.68, 41–50 (1995).
Hsiung, C. C.: A first course in differential geometry. New York: John Wiley, 1981.
Nielson, G. M.: A method for interpolating scattered data based upon a minimum norm network. Math. Comp.40, 253–271 (1983).
Powell, M. J. D., Sabin, M. A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Software3, 316–325 (1977).
Quak, E., Schumaker, L. L.: Cubic spline fitting using data dependent triangulations. Comput. Aided Geom. Design7, 293–301 (1990).
Renka, R. J., Cline, A. K.: A triangle-basedC 1 interpolation method. Rocky Mountain J. Math.14, 223–237 (1984).
Risler, J. J.: Mathematical methods for CAD. Cambridge: Cambridge University Press, 1992.
Schumaker, L. L.: Triangulation methods. In: Topics in multivariate approximation, (Schumaker, L. L., Chui, C. K., Utreras, F., eds.), pp. 219–273. New York: Academic Press, 1987.
Stead, S. E.: Estimation of gradients from scattered data. Rocky Mountain J. Math.14, 265–279 (1984).
Ženišek, A.: Interpolation polynomials on the triangle. Numer. Math.15, 283–296 (1970).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
De Marchi, S. On computing derivatives forC 1 interpolating schemes: an optimization. Computing 60, 29–53 (1998). https://doi.org/10.1007/BF02684328
Issue Date:
DOI: https://doi.org/10.1007/BF02684328