A refinement of the variation diminishing property of Bézier curves

Rachid Ait-Haddou; Taishin Nomura; Luc Biard

doi:10.1016/j.cagd.2009.12.001

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 journal › Article › peer-review

3 Scopus citations

Abstract

For a given polynomial F (t) = ∑_{i = 0}ⁿ p_i B_iⁿ (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 = k}^l p_i B_{i - k}^{l - k} (t) (counting multiplicities) with the number of sign changes in the two sequences (p₀, ..., p_k) and (p_l, ..., p_n) 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 language	English (US)
Pages (from-to)	202-211
Number of pages	10
Journal	Computer Aided Geometric Design
Volume	27
Issue number	2
DOIs	https://doi.org/10.1016/j.cagd.2009.12.001
State	Published - Feb 2010
Externally published	Yes

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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10.1016/j.cagd.2009.12.001

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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 B{\'e}zier curves.",

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AU - Ait-Haddou, Rachid

AU - Nomura, Taishin

AU - Biard, Luc

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AB - 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.

KW - Blossoming

KW - Bézier curve

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A refinement of the variation diminishing property of Bézier curves

Abstract

Keywords

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