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Tchebycheff Curves

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Total Positivity and Its Applications

Part of the book series: Mathematics and Its Applications ((MAIA,volume 359))

Abstract

Given an extended Tchebycheff space of dimension m + 1, we associate with it a normal curve C in real projective m—space P m. The curve C has the properties that any hyperplane intersects it in at most m points and that through any point of P m there pass at most m of its osculating hyperplanes. This allows us to study extended Tchebycheff spaces and Tchebycheffian splines in a geometric way. We introduce a generalized blossom which directly leads to generalizations of Bézier und B—spline curves. Furthermore, we obtain connections to isotropic curve theory and to recent results of Carnicer and Peña on B—bases and extensibility of Tchebycheff systems. The paper both surveys recent research and presents new results.

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Author information

Authors and Affiliations

  1. Université Joseph Fourier, LMC-IMAG, B.P. 53X, F-38041, Grenoble, France

    Marie-Laurence Mazure

  2. Institut für Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10, A-1040, Wien, Austria

    Helmut Pottnann

Authors
  1. Marie-Laurence Mazure
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  2. Helmut Pottnann
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Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, University of Zaragoza, Zaragoza, Spain

    Mariano Gasca

  2. IBM, T.J. Watson Research Center, Yorktown Heights, New York, USA

    Charles A. Micchelli

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© 1996 Springer Science+Business Media Dordrecht

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Mazure, ML., Pottnann, H. (1996). Tchebycheff Curves. In: Gasca, M., Micchelli, C.A. (eds) Total Positivity and Its Applications. Mathematics and Its Applications, vol 359. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8674-0_10

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  • DOI: https://doi.org/10.1007/978-94-015-8674-0_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4667-3

  • Online ISBN: 978-94-015-8674-0

  • eBook Packages: Springer Book Archive

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