Extension of Bézier curves of the same degree and parameter selection

Lanlan Yan; Zhiyong Rao; Tao Huang

doi:10.11834/jig.180005

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Extension of Bézier curves of the same degree and parameter selection
Vol. 23, Issue 9, Pages: 1411-1423(2018)
Received：22 January 2018，

Revised：2018-2-27，

Published：16 September 2018
DOI： 10.11834/jig.180005
稿件说明：

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DOI：

Lanlan Yan, Zhiyong Rao, Tao Huang. Extension of Bézier curves of the same degree and parameter selection[J]. Journal of Image and Graphics, 2018, 23(9): 1411-1423. DOI： 10.11834/jig.180005.

摘要

目的

本文旨在构造一种含形状参数的Bézier曲线，要求该曲线定义在代数多项式空间上，其基函数的次数与相同数量控制顶点所需Bernstein基函数的次数相同，对基函数以及相应曲线的计算要尽可能简单，并且要给出常见设计要求下曲线中形状参数的选取方案。

方法

以三次Bézier曲线为初始研究对象，依据由可调控制顶点定义可调曲线的思想，在两个内控制顶点中引入参数，与Bernstein基函数作线性组合生成形状可调曲线，再将曲线表达式改写成固定控制顶点与含参数的调配函数的线性组合，从而得出三次Bernstein基函数的含参数扩展基，借助递推公式得出更高次的含参数扩展基，然后观察基函数表达式的规律，给出所有含参数扩展基统一的显示表达式，分析了扩展基的性质，并由之定义含参数的曲线，分析了曲线的性质，给出了曲线的几何作图法以及光滑拼接条件，以曲线拉伸能量、弯曲能量、扭曲能量近似最小为目标，推导了曲线中形状参数的计算公式，再通过曲线图和曲率图对比分析了不同能量目标所得曲线的差异。

结果

由于所给含参数的扩展基并未提升Bernstein基函数的次数，且具有统一的显示表达式，因此本文方法在赋予Bézier曲线形状调整能力的同时并未增加计算量，由于提供了可以直接使用的形状参数的计算公式，因此在使用该方法时，符合设计要求的形状参数的确定变得简单，数值实例直观显示了所给曲线造型方法以及曲线中形状参数选取方案的正确性与有效性，体现了本文方法较文献中类似方法的优越之处。

结论

所给含参数扩展基的构造方法以及形状参数的选取方法具有一般性，该方法可以推广至构造含形状参数的三角域Bézier曲面。

Abstract

Objective

The purpose of this paper is to construct a type of Bézier curve with a shape parameter. We require the curves defined in algebraic polynomial space. The degree of the basis functions should be the same as the Bernstein basis functions

which needed the same number of control points. The calculation of the basis functions and corresponding curves should be as simple as possible. The selection scheme under common design requirements of the shape parameter in the curves should be provided.

Method

With the cubic Bézier curve as the initial research object and in accordance with the idea of defining a shape-adjustable curve by using adjustable control points

we introduce a parameter into the two inner control points. Let the control points with the parameter have a linear combination with the Bernstein basis functions to generate the shape adjustable curves. By rewriting the expression of the curves as the linear combination of the fixed control points and the blending functions with the parameter

we obtain the extended basis with the parameter of the cubic Bernstein basis functions. By using the recursive formula

we obtain the extended basis with the parameter of a high degree. Then

we observe the rule of the basis function expression and provide the uniform explicit expression of all extended basis functions with parameters. The properties of the extended basis functions are analyzed

and the corresponding curves with parameters are defined. The properties of the curves are analyzed. The geometric drawing method and smooth joining conditions of the curves are also provided. The calculation formula of the parameter

which causes the stretch

strain

and jerk energies of the curves to be approximately minimum

is deduced. The difference of the curves determined by different energy targets is compared and analyzed by using the graph of the curves and their curvatures.

Result

The method provides the Bézier curve shape adjustability without increasing the calculation amount due to the fact that the extended basis functions have the same degree as the Bernstein basis functions and have a uniform explicit expression. Determining the shape parameter that conforms to the design requirements when using this method is easy because the calculation formula of the shape parameter can be used directly. The numerical examples intuitively show the correctness and validity of the proposed curve modeling method and the shape parameter selection scheme in the curve. The illustration also shows the superiority of the method provided in this paper over similar methods presented in the literature.

Conclusion

The method of constructing an extended basis with the parameter and selection method of the shape parameter are general. This method can be extended to construct a triangular Bézier surface with parameter.

关键词

Keywords

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