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Monotonicity preservation on triangles

Authors: Michael S. Floater and J. M. Peña
Journal: Math. Comp. 69 (2000), 1505-1519
MSC (1991): Primary 41A10, 65D17; Secondary 41A63
Published electronically: May 20, 1999
MathSciNet review: 1677482
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Abstract: It is well known that Bernstein polynomials on triangles preserve monotonicity. In this paper we define and study three kinds of monotonicity preservation of systems of bivariate functions on a triangle. We characterize and compare several of these systems and derive some geometric applications.

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  • [1] J. M. Carnicer, M. García-Esnaola, and J. M. Peña, Convexity of rational curves and total positivity, J. Comp. Appl. Math. 71 (1996), 365-382. MR 97k:65043
  • [2] J. M. Carnicer and J. M. Peña, Shape preserving representations and optimality of the Bernstein basis, Adv. Comput. Math. 1 (1993), 173-196. MR 94i:65138
  • [3] J. M. Carnicer and J. M. Peña, Monotonicity preserving representations, Curves and Surfaces in Geometric Design, II, P. J. Laurent, A. Le Méhauté and L. L. Schumaker, (eds.), AKPeters, Boston, 1994, pp. 83-90. MR 95g:65198
  • [4] J. M. Carnicer and J. M. Peña, Total positivity and optimal bases, Total Positivity and its Applications (M. Gasca and C.A. Micchelli, eds.), Kluwer Academic Press, Dordrecht, 1996, pp. 133-155. MR 97i:41012
  • [5] G. Farin, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, Boston, 1988. MR 90c:65014
  • [6] R. T. Farouki and T. N. T. Goodman, On the optimal stability of the Bernstein basis, Math. Comp. 65 (1996), 1553-1566. MR 97a:65021
  • [7] T. N. T. Goodman, Variation diminishing properties of Bernstein polynomials on triangles, J. Approx. Theory 50 (1987), 111-126. MR 88g:41006
  • [8] T. N. T. Goodman, Further variation diminishing properties of Bernstein polynomials on triangles, Constr. Approx. 3 (1987), 297-305. MR 88j:41022
  • [9] T. N. T. Goodman, Shape preserving representations, Mathematical Methods in Computer Aided Geometric Design T. Lyche and L.L. Schumaker (eds.), Academic Press, New York, 1989, pp. 333-351. MR 91a:65031
  • [10] J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, A. K. Peters, Wellesley, MA, 1993. MR 94i:65003
  • [11] S. Karlin, Total Positivity, Stanford University Press, Stanford, 1968. MR 37:5667
  • [12] T. Sauer, Multivariate Bernstein polynomials and convexity, Comput. Aided Geom. Design 8 (1991), 465-478. MR 93a:41012

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Additional Information

Michael S. Floater
Affiliation: SINTEF Applied Mathematics, P.O. Box 124 Blindern, 0314 Oslo, NORWAY

J. M. Peña
Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, Edificio de Mate- máticas, Planta 1a, 50009 Zaragoza, SPAIN

Keywords: Monotonicity, shape preservation, bivariate Bernstein polynomials, control net
Received by editor(s): May 27, 1997
Received by editor(s) in revised form: December 7, 1998
Published electronically: May 20, 1999
Additional Notes: The authors were supported in part by the EU project CHRX-CT94-0522.
Article copyright: © Copyright 2000 American Mathematical Society

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