A refinement of the variation diminishing property of Bézier curves

Rachid Ait-Haddou, Taishin Nomura, Luc Biard*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

For a given polynomial F (t) = ∑i = 0n pi Bin (t), expressed in the Bernstein basis over an interval [a, b], we prove that the number of real roots of F (t) in [a, b], counting multiplicities, does not exceed the sum of the number of real roots in [a, b] of the polynomial G (t) = ∑i = kl pi Bi - kl - k (t) (counting multiplicities) with the number of sign changes in the two sequences (p0, ..., pk) and (pl, ..., pn) for any value k, l with 0 ≤ k ≤ l ≤ n. As a by product of this result, we give new refinements of the classical variation diminishing property of Bézier curves.

Original languageEnglish (US)
Pages (from-to)202-211
Number of pages10
JournalComputer Aided Geometric Design
Volume27
Issue number2
DOIs
StatePublished - Feb 1 2010

Keywords

  • Blossoming
  • Bézier curve
  • Polar derivative
  • Variation diminishing property

ASJC Scopus subject areas

  • Modeling and Simulation
  • Automotive Engineering
  • Aerospace Engineering
  • Computer Graphics and Computer-Aided Design

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