Degree reduction of interval Bézier curves☆
Introduction
In the past decade, there has been considerable interest in approximating curves and surfaces that arise in CAD applications by using curves and surfaces of lower degrees or simpler functional forms. The motivation for this research is the practical need to communicate product data between diverse CAD/CAM systems that impose fundamentally incompatible constraints on their representation schemes. For example, some systems restrict themselves to polynomial forms or limit the polynomial degrees that they accommodate.
However, almost all the approximation schemes guarantee only that an approximation will satisfy the prescribed tolerance, and none proposed to carry the detailed information on the approximation error forward to the subsequent applications. Such information can be of crucial importance in geometrical operations in other systems. To overcome this difficulty, Sederberg et al. [1] introduced a new representation form of parametric curves—the interval Bézier curves—that can transfer a complete description of approximation errors along with the curves to applications in other systems.
Inspired by Sederberg's work, Hu et al. [2], [3], [4], [5] and Tuohy et al. [6] recently turned to interval forms of geometric objects and rounded interval arithmetic to deal with the problem of lack of robustness—a problem that exists in all state-of-the-art solid modeling systems. In a floating point environment, representations of geometric objects are inaccurate, and geometrical computations are approximate. The consequences of this inaccuracy are (1) unreliability in geometrical computation and interrogation, such as boundary evaluation; and (2) inconsistency between the geometric and topology of geometric objects. The problem of lack of robustness in solid models has been clearly identified since the late 1980s [7] and remains an active topic of research. The series of works by Hu et al. indicate that using interval arithmetic will substantially increase the numerical stability in geometric computations, and thus enhance the robustness of current CAD/CAM systems.
In this paper, we discuss the problem of bounding interval Bézier curves with lower degree interval Bézier curves. Approximating Bézier curves with lower degree curves has been a hot spot in CAGD in the past decade, and a lot of research has focused on this problem [8], [9], [10], [11], [12], [13]. However, this research is concerned with how good the approximation is; none of it deals with approximation errors, or in other words, none has ever considered approximating interval Bézier curves with lower degree interval Bézier curves. As far as the authors are aware, the only existing work relating to degree reduction of interval Bézier curves is Rokne's paper [14], in which the author used (the graph of) an interval polynomial to bound (the graph of) a higher degree interval polynomial. In this paper, we present two totally different algorithms—Linear Programming and Optimal Approximation—to solve the same problem, and apply the algorithms to the problem of bounding an interval Bézier curve with a lower degree interval Bézier curve.
The organization of this paper is as follows. We first briefly review the definitions of interval arithmetic and interval Bézier curves. Then we describe two algorithms to bound an interval polynomial with a lower degree interval polynomial. The interval polynomial bounding algorithms lead directly to methods of bounding interval Bézier curves with lower degree interval Bézier curves. Finally, in the last section we compare the above two methods with the previous method through illustration of examples. The examples show that the two methods we present in this paper produce much better bounds than the previous method does, with almost the same computational costs.
Section snippets
Interval arithmetic and interval Bézier curves
An interval is the set of real numbers defined byThe interval [a,b] is said to be degenerate if a=b. Two intervals, [a,b] and [c,d], are said to be equal if a=c and b=d. The width of an interval [a,b] is b−a, and the absolute value is |[a,b]|=max(|a|,|b|). Sometimes we use a convenient shorthand notation for intervals, denoting them by a single symbol enclosed in square parentheses, for example, [u]=[a,b] and [v]=[c,d].
Given two intervals, [a,b] and [c,d], the interval
Degree reduction of interval polynomials
In this section, we develop algorithms to bound an interval polynomial with a lower degree interval polynomial. We wish to make the bound as tight as possible. Problem 1 Given an interval polynomial [p](t) of degree n,find an interval polynomial [q](t) of degree m<n,such thatand the width of [q](t) is as small as possible. We call [q](t) an interval polynomial bound of [p](t); qmax and qmin are upper bound and lower bound
Degree reduction of interval Bézier curves
In this section, we apply the algorithms for the degree reduction of interval polynomials to the degree reduction of interval Bézier curves. The key observation is the following: Theorem 4 Given an internal Bézier curve of degree n:Ifandare degree m<n polynomial bounds of [x](t) and [y](t), respectively, then interval Bézier curvebounds
Examples and comparisons
In this section, we compare the three different interval curve degree reduction methods—Rokne's method (RM), Linear Programming (LPM) and Optimal Approximation (OAM)—through illustration of examples. These examples demonstrate that the methods presented in this paper are much superior to Rokne's method, and that the Optimal Approximation method yields the best approximation results. Example 1 We consider a degree six polynomial whose magnitude drastically changes on [0,1]:
Conclusions
In this paper, we put forward the problem of how to bound an interval Bézier curve with a lower degree interval Bézier curve. This problem is crucial in transferring errors to subsequent applications in solid modeling systems. We proposed two methods—Linear Programming and Optimal Approximation—to solve this problem. Theoretical results and examples show that these two approaches are much superior to the previous method with about the same computational costs, and that the Optimal Approximation
Dr Falai Chen is currently a professor in the Mathematical Department at the University of Science and Technology of China. He received his BS (1987), MS (1989) and PhD degrees (1994) from the University of Science and Technology of China. He visited the Computer Graphics Lab at Brigham Young University, USA, from April 1994 to August 1995; and he also visited the CAD/CAM center at the Hong Kong University of Science and Technology for six months in 1997. His research interests include Computer
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Dr Falai Chen is currently a professor in the Mathematical Department at the University of Science and Technology of China. He received his BS (1987), MS (1989) and PhD degrees (1994) from the University of Science and Technology of China. He visited the Computer Graphics Lab at Brigham Young University, USA, from April 1994 to August 1995; and he also visited the CAD/CAM center at the Hong Kong University of Science and Technology for six months in 1997. His research interests include Computer Aided Geometric Design and Computer Graphics.
Mr Wenping Lou is currently a graduate student in the Mathematical Department at the University of Science and Technology of China. He received his BS degree from the University of Science and Technology of China. His research interests include Computer Aided Geometric Design and Computer Graphics.
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Supported by National Natural Science Foundation of China and Science Foundation of State Educational Commission of China.