scholarly journals Modified Bézier Curves with Shape-Preserving Characteristics Using Differential Evolution Optimization Algorithm

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Mohammad Asif Zaman ◽  
Shuvro Chowdhury
Keyword(s):  
Differential Evolution ◽  
Optimization Algorithm ◽  
Shape Control ◽  
Parametric Equation ◽  
Shape Preserving ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Data Points ◽  
Cost Parameters ◽  
Differential Evolution Optimization Algorithm

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.

scholarly journals Shape Modification forλ-Bézier Curves Based on Constrained Optimization of Position and Tangent Vector

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Gang Hu ◽  
Xiaomin Ji ◽  
Xinqiang Qin ◽  
Suxia Zhang
Keyword(s):  
Constrained Optimization ◽  
Shape Control ◽  
Control Parameter ◽  
Tangent Vector ◽  
Control Points ◽  
Shape Modification ◽  
Shape Preserving ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curve Design

Besides inheriting the properties of classical Bézier curves of degreen, the correspondingλ-Bézier curves have a good performance on adjusting their shapes by changing shape control parameter. Specially, in the case where the shape control parameter equals zero, theλ-Bézier curves degenerate to the classical Bézier curves. In this paper, the shape modification ofλ-Bézier curves by constrained optimization of position and tangent vector is investigated. The definition and properties ofλ-Bézier curves are given in detail, and the shape modification is implemented by optimizing perturbations of control points. At the same time, the explicit formulas of modifying control points and shape parameter are obtained by Lagrange multiplier method. Using this algorithm,λ-Bézier curves are modified to satisfy the specified constraints of position and tangent vector, meanwhile the shape-preserving property is still retained. In order to illustrate its ability on adjusting the shape ofλ-Bézier curves, some curve design applications are discussed, which show that the proposed method is effective and easy to implement.

Smoothness of C-Bezier Curves

2000 ◽  
Author(s):  
Manhong Wen ◽  
Kwun-Lon Ting
Keyword(s):  
Shape Control ◽  
Control Point ◽  
Design Procedure ◽  
Design Parameters ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Point Distribution ◽  
G1 Continuity ◽  
G2 Continuity

Abstract This paper presents G1 and G2 continuity conditions of c-Bezier curves. It shows that the collinear condition for G1 continuity of Bezier curves is generally no longer necessary for c-Bezier curves. Such a relaxation of constraints on control points is beneficial from the structure of c-Bezier curves. By using vector weights, each control point has two extra free design parameters, which offer the probability of obtaining G1 and G2 continuity by only adjusting the weights if the control points are properly distributed. The enlargement of control point distribution region greatly simplifies the design procedure to and enhances the shape control on constructing composite curves.

scholarly journals Approximate Multidegree Reduction ofλ-Bézier Curves

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Gang Hu ◽  
Huanxin Cao ◽  
Suxia Zhang
Keyword(s):  
Shape Control ◽  
Control Parameter ◽  
Linear Equations ◽  
Least Square ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Degree Reduction ◽  
Bezier Curves ◽  
Bézier Curves

Besides inheriting the properties of classical Bézier curves of degreen, the correspondingλ-Bézier curves have a good performance in adjusting their shapes by changing shape control parameter. In this paper, we derive an approximation algorithm for multidegree reduction ofλ-Bézier curves in theL2-norm. By analysing the properties ofλ-Bézier curves of degreen, a method which can deal with approximatingλ-Bézier curve of degreen+1byλ-Bézier curve of degreem  (m≤n)is presented. Then, in unrestricted andC0,C1constraint conditions, the new control points of approximatingλ-Bézier curve can be obtained by solving linear equations, which can minimize the least square error between the approximating curves and the original ones. Finally, several numerical examples of degree reduction are given and the errors are computed in three conditions. The results indicate that the proposed method is effective and easy to implement.

Sign in / Sign up

Export Citation Format

Share Document