دانلود رایگان مقاله تقریب یکنواخت از یک دایره با یک منحنی چند جمله ای پارامتری

1. Introduction

A circle arc is a basic object in CAGD and in many applications. Conics are the oldest curves, and are used in architecture, robotics, and in many other fields. The unit circle has a nice parameterization c(t) = (cos t, sint), t ∈ [0, 2π). In CAGD, parametric polynomial and rational curves and splines are fundamental objects. A circle arc does not have a parametric polynomial representation, however, it can be represented by using a quadratic rational Bézier curve. The whole circle can thus be represented by a quadratic rational spline, e.g. as a NURBS. A natural question is, whether it is possible to obtain a good parametric polynomial approximation of a circular arc. A lot of papers study good approximation of circular segments with the radial error as the parametric distance. Quadratic Bézier approximants are considered in Mørken (1991), and their generalizations to the cubic case can be found in Dokken et al. (1990) and Goldapp (1991). The quartic case is systematically studied in Ahn and Kim (1997), Kim and Ahn (2007) and Hur and Kim (2011), and quintic Bézier approximants are derived in Fang (1998, 1999). Recently, quartic G1 approximants were analyzed in Kovacˇ and Žagar (2014). General results on Hermite type approximation of conic sections by parametric polynomial curves of odd degree are given in Floater (1995, 1997). The results hold true only asymptotically, i.e., for small segments of a particular conic section. Hermite approximation of ellipse segments by cubic parametric Bézier curves is studied in Dokken (2003) and also in Dokken (1997). In recent years some quite surprisingly good approximations of the whole circle were obtained. An approach based on Taylor approximation was improved by idea of geometric interpolation and a construction of polynomial approximants for all odd degrees was obtained in Lyche and Mørken (1994). The construction that covered also even degrees was presented in Jaklicˇ et al. (2007a). By looking at the problem from a different perspective, it turned out that the obtained construction was just one of several solutions of a nonlinear problem, and that there exist better solutions. In Jaklicˇ et al. (2013), the best such solution was presented, which gives a good approximation of a conic section. It has many nice properties, it is symmetric, shape preserving, it gives a high order approximation of the whole circle c, and for higher degrees it circles the circle several times before it deviates from c. Furthermore, it is given in a closed form.