带局部形状参数的代数三角样条曲线曲面的构造

Construction of Algebraic and Trigonometric Spline Curves and Surfaces with Local Shape Parametric

在计算机辅助几何设计中,带形状参数三角B样条曲线已经是个热点问题,然而通常定义的带形状参数的曲线曲面中的参数是全局性的,不能局部调整曲线曲面的形状且是C^2连续,为了更好地控制和调整曲线曲面,构造了带局部形状参数的代数三角样条曲线曲面,简称为AT-spline.同时讨论了该曲线曲面的一些重要的性质,这种曲线不仅具有三角多项式的性质,同时具有局部可调性,可很好的表示曲线曲面.当-2≤λ_i,μ_i≤1时,带参数的AT-Spline曲线满足G^1连续,如果两个相邻曲线中的参数μ_i=λ_(i+1)或μ_i=λi=μ_(i+1)=λ_(i+1)时,则带参数的AT-Spline曲线C_4(λ_i,μ_i;u)满足C^1∩G^2连续.同时还构造了旋转面,讨论了形状参数对旋转面的外形的影响并给出了实例,从实验结果来看,显示了该方法的有效性.

In computer aided geometric design, the construction of trigonometric B-spline curves with shape parameters has become the hotspot. However,the shape parameters of the curves and surfaces in previous papers are all global parameters. In order to provide more flexible approaches for designers,the algebraic and trigonometric Spline curves and surfaces are constructed as a generalization of the traditional cubic uniform B-spline curves and surfaces. Possessing multiple local shape control parameters, AT-spline curves and surfaces not only inberit the properties of cubic uniform B-spline curves,but also exhibit better performance when adjusting its local shapes through two shape control parameters. When-2≤λi,μi≤1, an AT-spline curve C4 （λi,μi;u）satisfies G^1 continuity.If two adjacent AT-spline curves segments Ci,4 （ λi,μi;t） and Ci ＋ 1.4 （ λi＋1,μi ＋1;t ） are obtained by the condition μi= λi＋1 or μi= λi =μi ＋1 = λi＋1 = λ, the AT-spline curves C4 （ λi ,μi ; u） possesses C1 ∩ G^2 continuity. Particularly, to adjust and control the shapes of rotational surfaces more elegantly,the AT-spline rotational surfaces with two local shape parameters are presented and utilized.