Bézier曲线的同次扩展及其参数选择

Extension of Bézier curves of the same degree and the parameter selection

Yan Lanlan,Rao Zhiyong,Huang Tao()

Abstract
Abstract: Objective The purpose of this paper is to construct a kind of Bézier curves with 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 by the same number of control points. The calculation of the basis functions and the corresponding curves should be as simple as possible. The selection scheme under common design requirements of the shape parameter in the curves should be given. Method With the cubic Bézier curve as the initial research object, and according to the idea of defining shape adjustable curve by adjustable control points, we introduce parameter into the two inner control points. Let the control points with parameter have a linear combination with the Bernstein basis functions to generate the shape adjustable curves. Rewriting the expression of the curves as the linear combination of the fixed control points and the blending functions with parameter, then we obtain the extended basis with parameter of the cubic Bernstein basis functions. By using the recursive formula, we obtain the extended basis with parameter of higher degree. Then, we observe the rule of the basis functions expression, and give the uniform explicit expression of all extended basis functions with parameter. The properties of the extended basis functions are analyzed, and the corresponding curves with parameter are defined. The properties of the curves are analyzed. The geometric drawing method and the smooth joining conditions of the curves are also given. The calculation formula of the parameter which makes the stretch energy, strain energy and jerk energy of the curves approximate minimum is deduced. The difference of the curves determined by different energy targets is compared and analyzed by graph of the curves and their curvatures. Result Due to the fact that the extended basis functions have the same degree as the Bernstein basis functions and they have the uniform explicit expression, the method gives the Bézier curves shape adjustability without increasing the calculation amount. Because of the calculation formula of the shape parameter can be used directly, it is easy to determine the shape parameter that conforms to the design requirements when using this method. 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 given in this paper over the similar methods given in literature. Conclusion The method of constructing extended basis with parameter and the selection method of shape parameter are general. This method can be extended to construct triangular Bézier surface with parameter.
Keywords
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