دانلود رایگان مقاله کلیت چندجمله ای برنشتاین و منحنی Bezier بر اساس حساب آمبرال

Abstract

The investigation of the umbral calculus based generalization of Bernstein polynomials and Bézier curves is continued in this paper: First a generalization of the de Casteljau algorithm that uses umbral shift operators is described. Then it is shown that the quite involved umbral shifts can be replaced by a surprisingly simple recursion which in turn can be understood in geometrical terms as an extension of the de Casteljau interpolation scheme. Namely, instead of using only the control points of level r−1 to generate the points on level r as in the ordinary de Casteljau algorithm, one uses also points on level r−2 or more previous levels. Thus the unintuitive parameters in the algebraic definition of generalized Bernstein polynomials get geometric meaning. On this basis a new direct method for the design of Bézier curves is described that allows to adapt the control polygon as a whole by moving a point of the associated Bézier curve.

5. A new direct method for the design of Bézier curves

Usually the designer of a Bézier curve sets up in an intuitive way a control polygon and inspects the resulting Bézier curve. If the curve is not as desired, then the control points are moved interactively one by one, until the result is satisfying. In the past there have also been invented a number of direct methods for curve design, that allow to change the whole Bézier curve by picking a point on the curve and changing its geometric constraints, e.g., its position in ambient space, the tangent direction and magnitude, or the magnitude of the curvature at the point (see for example Bartels and Beatty, 1989; Fowler and Bartels, 1993; Gleicher, 1992). The purpose of the present section is to indicate, how the additional freedom gained by the parameters a¯2,a¯3,... can be used to extend the known direct methods. Note that by the generalized de Casteljau algorithm of Theorem 4.6 these parameters do not describe local differential–geometric properties, but global properties of the generalized Bézier curves, e.g., an a¯2 > 0 and an a¯2 < 0 leads to a global decrease resp. increase (cf. Fig. 4) of curvature compared to the ordinary Bézier curves. We hope that other researchers can apply this approach to their particular problem. The new direct method is described and illustrated first from the perspective of the designer, then we discuss the mathematics behind the working steps. (Note, that also a weighting of control points can be done in the usual way in the generalized setup.)