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.
Bézier curve is a curve fitting tool for constructing freeform smooth parametric curves. Bézier curves are widely used in computer aided geometry design, data structure modelling, mesh generating techniques, and computer graphics applications [
A standard curve fitting problem is defined by a set of raw data points, referred to as
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Comparison of interpolated curve and Bézier curve for test function
Although Bézier curves usually follow the general trend of the control polygon, it often underestimates the slope of the control polygon. Due to inherit smoothing characteristics, it fails to follow control polygons with sharply varying sides. Also, Bézier curves lack flexibility of controlling the shape of the curve. It is not possible to generate multiple Bézier curves with different shapes for the same control polygon. This flexibility is often desired to finetune and optimize a curvefitting problem. For this reason, modified versions of Bézier curves with shaping parameters have been developed [
DE algorithm was first developed by Storn and Price in 1997 [
The paper is organized as follows. Section
For a set of
To incorporate additional parameters in the Bézier curve equation, a few constraints must be presatisfied. It is required that the modified Bézier curve spans the same region as the conventional Bézier curve, which is limited by the minimum and maximum value of the
After defining the modified Bézier curve equations, it becomes necessary to mathematically define the
The slope of the modified curve can be calculated from using the chain rule:
To define the optimality of the curve, a
Although this paper concentrates on planar Bézier curves, the proposed method can be extended for general Bézier curves. The planar curves are defined by (
DE is a population based heuristic evolutionary optimization algorithm [
For the curve fitting problem discussed in this paper, the solution vectors represent a set of values of the shaping parameters,
The solution space must be limited by defining possible range of values of
The proposed method is implemented and tested using computer coding. For simulation, the weight factors of (
For testing purposes, the proposed modified Bézier curve is used to fit four test functions. The performance of the modified curve is measured in terms of sum of squares error between the curve and control polygon,
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Table for test function,






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Comparison of modified Bézier curve and conventional Bézier curve for test function
Comparison of modified Bézier curve and conventional Bézier curve for test function
Comparison of modified Bézier curve and conventional Bézier curve for test function
Comparison of modified Bézier curve and conventional Bézier curve for test function
To ensure that the DE algorithm reached convergence, the cost versus generation number plot was observed for all the test functions. All the plots show a decreasing nature in the beginning and flats out around iteration 110 to 160. The flat values of average cost and minimum cost indicate that saturation is reached [
Cost values versus iteration number for test function
A modified parametric equation is developed by modifying the equation of Bézier curve. The resulting modified curve allows control over the shape of the fitted curve by introducing shaping parameters. Using DE algorithm, the optimum value of the shaping parameters can be found. The optimum curve matches the shape of the control polygon and the slope of the control polygon with a much higher degree of accuracy compared to the conventional Bézier curve. Multiple test functions are used to test the proposed method. A number of performance parameters are defined, which shows that the proposed method outperforms conventional Bézier curve. It is found that for sharply varying data points, the proposed method produces a curve that follows the control polygon more closely compared to conventional Bézier curve. As the proposed method is general, it can be used on any discrete sets of data of points to produce a highly accurate shape preserving curve.