BERNSTEIN-BEZIER METHODS FOR COMPUTER-AIDED-DESIGN OF FREE-FORM CURVES AND SURFACES
BERNSTEIN-BEZIER METHODS FOR COMPUTER-AIDED-DESIGN OF FREE-FORM CURVES AND SURFACES
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DOI：
10.1145/321812.321824
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发表时间：
1974-01-01
影响因子：
2.5
通讯作者：
RIESENFELD, RF
中科院分区：
文献类型：
--
作者：
GORDON, WJ;RIESENFELD, RF
定义在[0，1]上的函数φ的三次伯恩斯坦多项式逼近为φ mμ=0 <$（μ/m）φμ（s），其中权φμ（s）是二项密度函数.伯恩斯坦近似继承了许多全局特性的双曲型，如单调性和凸性，他们总是至少“光滑“的双曲型，其中“光滑“是指起伏的数量，总变差，和可微类双曲型。历史上，他们的相对缓慢的收敛在L∞-范数往往不利于他们在实际应用中的使用。然而，在一大类问题中，近似函数的光滑性比拟合的接近性更重要。这是特别真实的与计算机辅助几何设计的曲线和曲面的美学标准和形状的内在属性是主要考虑的问题。对于后一类问题，雷诺的P·贝塞尔成功地利用了参数伯恩斯坦多项式的性质。本文的目的是分析贝塞尔技术，并探讨各种扩展和推广。在续集中，作者认为这里所包含的结果的扩展到自由形式的曲线和曲面设计使用polynomialsplines。这些B样条方法有几个优点，在本文件中描述的技术。
Themth degree Bernstein polynomial approximation to a function ƒ defined over [0, 1] is Σmμ=0 ƒ(μ/m)φμ(s), where the weightsφμ(s) are binomial density functions. The Bernstein approximations inherit many of theglobalcharacteristics of ƒ, like monotonicity and convexity, and they always are at least as“smooth”as ƒ, where “smooth” refers to the number of undulations, the total variation, and the differentiability class of ƒ.Historically, their relatively slow convergence in the L∞-norm has tended to discourage their use in practical applications.However, in a large class of problems the smoothness of an approximating function is of greater importance than closeness of fit. This is especially true in connection with problems of computer-aidedgeometric designof curves and surfaces where aesthetic criteria and theintrinsic properties of shapeare major considerations. For this latter class of problems, P. Bézier of Renault has successfully exploited the properties of parametric Bernstein polynomials. The purpose of this paper is to analyze the Bézier techniques and to explore various extensions and generalizations. In a sequel, the authors consider the extension of the results contained herein to free-form curve and surface design using polynomialsplines. These B-spline methods have several advantages over the techniques described in the present paper.