Bases for shape preserving curves

Francesca Pitolli

doi:10.33993/jnaat321-738

Authors

Francesca Pitolli Universita di Roma “La Sapienza”, Italy

DOI:

https://doi.org/10.33993/jnaat321-738

Keywords:

total positivity, shape preserving, optimal bases

Abstract views: 174

Abstract

The shape preserving properties of a curve in \(\mathbb{R}^2\) depend on the properties of the function basis we use in its representation. Both sign consistent and totally positive bases have shape preserving properties useful in Computer Aided Geometric Design. Some of the most useful properties are lightened and some examples of shape preserving bases are given.

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References

Barsky, B. A., The beta-splines: a local representation based on shape parameters and fundamental geometric measures, Ph.D. dissertation, University of Utah, 1981.

Carnicer, J. M., Sign consistency and shape properties, in Mathematical Methods for Curves and Surfaces II, M. Daehlen, T. Lyche and L. L. Schumaker (eds.), Vanderbilt University Press, Nashville, pp. 41-48, 1998.

Carnicer, J. M., Goodman, T. N. T. and Peña, J. M., A generalization of the variation diminishing property, Adv. Comput. Math., 3, no. 4, pp. 375--394, 1995, https://doi.org/10.1007/BF02432004. DOI: https://doi.org/10.1007/BF02432004

Carnicer, J. M. and Peña, J. M., Shape preserving representation and optimality of the Bernstein basis, Adv. Comput. Math., 1, no. 2, pp. 173-196, 1993, https://doi.org/10.1007/BF02071384. DOI: https://doi.org/10.1007/BF02071384

Carnicer, J. M. and Peña, J. M., Totally positive bases for shape preserving curves design and optimality of B-splines, Comput. Aided Geom. Design, 11, no. 6, pp. 633-654, 1994, https://doi.org/10.1016/0167-8396(94)90056-6. DOI: https://doi.org/10.1016/0167-8396(94)90056-6

Carnicer, J. M. and Peña, J. M., Total positivity and optimal bases, in Total Positivity and its Applications (JACA 1994), M. Gasca and C. A. Micchelli (eds.), Kluwer Academic Publishers, Dordrecht, pp. 133--155, 1996. DOI: https://doi.org/10.1007/978-94-015-8674-0_8

Carnicer, J. M. and Peña, J. M., Characterization of the optimal Descartes' rules of signs, Math. Nachr., 189, pp. 33-48, 1998. DOI: https://doi.org/10.1002/mana.19981890104

de Casteljau, P., Outillage méthodes calcul, André Citröen Automobiles S.A., Paris, 1959.

Dyn, N. and Micchelli, C. A., Piecewise polynomial spaces and geometric continuity of curves, Numer. Math., 54, no. 3, pp. 319-337, 1988, https://doi.org/10.1007/BF01396765. DOI: https://doi.org/10.1007/BF01396765

Farin, G., Visually C² cubic splines, Comput. Aided Design, 14, pp. 137-139, 1982, https://doi.org/10.1016/0010-4485(82)90326-8. DOI: https://doi.org/10.1016/0010-4485(82)90326-8

Farin, G., Curves and Surfaces for Computer Aided Geometric Design, Academic Press, Boston, MA, 1988. DOI: https://doi.org/10.1016/B978-0-12-460515-2.50020-2

Gasca, M. and Micchelli, C. A., Total Positivity and Its Applications, Kluwer Academic Publishers, Dordrecht, 1996. DOI: https://doi.org/10.1007/978-94-015-8674-0

Goodman, T. N. T., Shape preserving representations, in Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. L. Schumaker (eds.), Academic Press, New York, pp. 333-357, 1989. DOI: https://doi.org/10.1016/B978-0-12-460515-2.50027-5

Goodman, T. N. T., Inflections on curves in two and three dimensions, Comput. Aided Geom. Design, 8, no. 1, pp. 37-50, 1991. DOI: https://doi.org/10.1016/0167-8396(91)90048-G

Goodman, T. N. T., Total positivity and the shape of curves, in Total Positivity and its Applications, M. Gasca and C. A. Micchelli (eds.), Kluwer Academic Publishers, pp. 157-186, 1996. DOI: https://doi.org/10.1007/978-94-015-8674-0_9

Goodman, T. N. T. and Micchelli, C. A., Corner cutting algorithms for the Bézier representation of free form curves, Linear Algebra Appl., 99, pp. 225-252, 1988. DOI: https://doi.org/10.1016/0024-3795(88)90134-6

Gori, L., Pezza, L. and Pitolli, F., A class of totally positive blending B-Bases, in Curve and Surface Design: Saint Malo 1999, P. J. Laurent, P. Sablonnière, L. L. Schumaker (eds.), Vanderbilt University Press, Nashville, TN, pp. 119-126, 2000.

Karlin, S., Total Positivity, Stanford University Press, Stanford, 1968.

Schmeltz, G., Variationesreduzierende Kurvendarstellungen und Krümmungskriterien für Bézierflächen, Thesis, Fachbereich Mathematik, Technische Hochschule Darmstadt, 1992.

Schumaker, L. L., Spline functions: basic theory, Krieger Publishing Company, Malabar, Florida, 1993.