A parametric equation for a modified Bézier 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 Bézier curves. From the obtained results, it is observed that the proposed method produces a curve that fits the data points more accurately. 1. Introduction Bézier curve is a curve fitting tool for constructing free-form smooth parametric curves. Bézier curves are widely used in computer aided geometry design, data structure modelling, mesh generating techniques, and computer graphics applications [1–3]. 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 Bézier 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
References
[1]
F. Curtis Gerald and O. Patrick Wheatley, Applied Numerical Analysis, Pearson, 7th edition.
[2]
K. Anastasiou and C. T. Chan, “Automatic triangular mesh generation scheme for curved surfaces,” Communications in Numerical Methods in Engineering, vol. 12, no. 3, pp. 197–208, 1996.
[3]
M. A. Scott, M. J. Borden, C. V. Verhoosel, T. W. Sederberg, and T. J. R. Hughes, “Isogeometric finite element data structures based on Bézier extraction of T-splines,” International Journal for Numerical Methods in Engineering, vol. 88, no. 2, pp. 126–156, 2011.
[4]
W. Wen-tao and W. Guo-zhao, “Bézier curves with shape parameter,” Journal of Zhejiang University Science A, vol. 6, no. 6, pp. 497–501, 2005.
[5]
Q.-B. Wu and F.-H. Xia, “Shape modification of Bézier curves by constrained optimization,” Journal of Zhejiang University Science, vol. 6, no. 1, pp. 124–127, 2005.
[6]
F. A. Sohel, G. C. Karmakar, and L. S. Dooley, “An improved shape descriptor using bezier curves,” in Pattern Recognition and Machine Intelligence, vol. 3776 of Lecture Notes in Computer Science, pp. 401–406, Springer, Berlin, Germany, 2005.
[7]
M. I. Grigor'cprimeev, V. N. Malozemov, and A. N. Sergeev, “Bernstein polynomials and composite Bézier curves,” Journal of Computational Mathematics and Mathematical Physics, vol. 46, no. 11, pp. 1962–1971, 2006.
[8]
G. T. Tachev, “Approximation of a continuous curve by its Bernstein-Bézier operator,” Mediterranean Journal of Mathematics, vol. 8, no. 3, pp. 369–381, 2011.
[9]
L. Yang and X.-M. Zeng, “Bézier curves and surfaces with shape parameters,” International Journal of Computer Mathematics, vol. 86, no. 7, pp. 1253–1263, 2009.
[10]
Q. Chen and G. Wang, “A class of Bézier-like curves,” Computer Aided Geometric Design, vol. 20, no. 1, pp. 29–39, 2003.
[11]
N. F. Fijasri, S. H. Yahaya, and J. M. Ali, “Bézier-like quartic curve with shape control,” in 2nd IMT-GT Regional Conference on Mathematics, Statistics and Applications, pp. 1–6, June 2006.
[12]
W.-Q. Shen and G.-Z. Wang, “A class of quasi Bézier curves based on hyperbolic polynomials,” Journal of Zhejiang University Science, vol. 6, no. 1, pp. 116–123, 2005.
[13]
？. Di？ibüyük and H. Oru？, “A generalization of rational Bernstein-Bézier curves,” BIT. Numerical Mathematics, vol. 47, no. 2, pp. 313–323, 2007.
[14]
R. Storn and K. Price, “Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces,” Journal of Global Optimization, vol. 11, no. 4, pp. 341–359, 1997.
[15]
K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution—A Practical Approach to Global Optimization, Springer, 2005.
[16]
P. Pandunata and S. M. H. Shamsuddin, “Differential evolution optimization for Bézier curve fitting,” in Proceedings of the 7th International Conference on Computer Graphics, Imaging and Visualization (CGIV '10), pp. 68–72, August 2010.
[17]
T. Rogalsky, “Bézier parameterization for optimal control by differential evolution,” in International Conference on Genetic and Evolutionary Methods, pp. 28–34, July 2011.
