Improved bounds on the magnitude of the derivative of rational Bézier curves
Introduction
A rational Bézier curve γ of degree n is given by the control points Pi ∈ Rd and the corresponding weights ωi ∈ R in the formwhere is the Bernstein polynomials given by . In this paper we assume that all the weights ωi are positive, which are sufficient for practical applications.
Derivative formula of a parametric curve has important applications in geometric design and applied mathematics [4]. The estimation of bounds on derivatives of rational Bézier curves has important applications in computer aided geometric design [3] and computer graphics [6]. Many authors gave bounds on the magnitude of the derivatives of rational Bézier curves [1], [2], [3], [5], [6], [7], [8]. In connection with this paper, let D = maxi{∥Pi+1 − Pi∥}, Floater [1] obtained the following derivative inequality:Recently, Selimovic [5] derived a new inequality:where .
Zhang and Ma [7] improved Selimovic’s result when the degree of the rational Bézier curves satisfy 2 ⩽ n ⩽ 7:where f(2) = f(3) = 1, f(4) = f(5) = f(6) = 2, f(7) = 3.
In the original papers [5], [7], W in (3), (4) takes the false definition W = max{ω, 1/ω}, ω = maxi{ωi/ωi+1}. The right definition is that . These errors are pointed and corrected by Huang and Su [3].
In this paper we give a new result which is written as follows: Theorem 1 For any rational Bézier curve defined by (1), we havewhere .
In Section 3 we will show that (5) is sharper than (3).
Section snippets
The proof of Theorem 1
We follow the approach of Floater [1], and exploit the representation of the derivative:
Here and are the intermediate weights and points of the de Casteljau algorithm. By setting and they are given by:
First we give the following Lemma. Lemma 1 For the intermediate weights of the kth step of the de Casteljau algorithm, we have: Proof According to (7) we have
Remarks and discussions
Since M ⩾ 0, we have that . Then, from Theorem 1, we deduce Corollary 1 For any rational Bézier curve defined by (1), we have
By [7], we have and using Theorem 1, we deduce Corollary 2 For any rational Bézier curve defined by (1), we have
Both (13), (14) are simple.
Now let us compare the different results discussed in this paper.
It is easy to see that (14) is always stronger than (3), therefore the bound (5) is also an improvement of (3).
We also observe
Acknowledgement
This work is supported by NSFC (61003194, 10926058, 11026107).
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