%0 Journal Article
%T Modified B¨¦zier Curves with Shape-Preserving Characteristics Using Differential Evolution Optimization Algorithm
%A Mohammad Asif Zaman
%A Shuvro Chowdhury
%J Advances in Numerical Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/858279
%X A parametric equation for a modified B¨¦zier curve is proposed for curve fitting applications. The proposed equation contains shaping parameters to adjust the shape of the fitted curve. This flexibility of shape control is expected to produce a curve which is capable of following any sets of discrete data points. A Differential Evolution (DE) optimization based technique is proposed to find the optimum value of these shaping parameters. The optimality of the fitted curve is defined in terms of some proposed cost parameters. These parameters are defined based on sum of squares errors. Numerical results are presented highlighting the effectiveness of the proposed curves compared with conventional B¨¦zier curves. From the obtained results, it is observed that the proposed method produces a curve that fits the data points more accurately. 1. Introduction B¨¦zier curve is a curve fitting tool for constructing free-form smooth parametric curves. B¨¦zier curves are widely used in computer aided geometry design, data structure modelling, mesh generating techniques, and computer graphics applications [1¨C3]. These curves are also used in different fields of mechanical and electrical engineering for modelling complex surface geometries [1]. Because of the wide field of applications, efficient techniques for improving and controlling the shape of B¨¦zier curves have become an important field of research [4, 5]. A standard curve fitting problem is defined by a set of raw data points, referred to as control points. The polygon obtained by connecting all the control points is termed as the control polygon. For most applications, it is required to find a free-form curve that most closely follows the control polygon. Conventional least-squares curve fitting techniques can fit a mathematical equation through the control points. However, this technique is not always applicable as the control points do not necessarily follow any standard mathematical equation models. Spline interpolation results in a continuous curve that matches the control polygon to a high degree [1], but the curve is expressed in terms of piece-wise defined functions. In many cases a single equation for the whole curve is required for mathematical operations. In such cases spline interpolations are not useful. In case of an interpolating polynomial, the interpolated curve goes through all the control points; however, the interpolated region between two control points often deviates significantly from the control polygon. This problem arises because the interpolating polynomial only tries to match the
%U http://www.hindawi.com/journals/ana/2013/858279/