基于几何连续的AT-β-Spline曲线曲面的构造

为了更好地修改给定的样条曲线曲面,构造了满足几何连续的带两类形状参数的代数三角多项式样条曲线曲面,简称为AT-β-Spline.这种代数三角曲线曲面不仅具有普通三角多项式的性质,而且具有全局的和局部的形状可调性.同时还具备较为灵活的连续性.当两类形状参数在给定的范围内任意取值时,这种带两类形状参数的AT-β-Spline曲线满足一阶几何连续性;如果给定两段相邻曲线段中的两类形状参数满足-1≤α≤1,μ_i=λ_(i+1)或μ_i=λ_i=μ_(i+1)=λ_(i+1)时,则带两类形状参数的AT-β-Spline曲线满足C~1∩G~2连续.另外利用奇异混合的思想,构造了满足C~1∩G~2插值AT-β-Spline曲线,解决曲线反求的几何连续性等问题.同时还给出了旋转面的构造,描述了两类形状参数对旋转面的几何外形的影响;当形状参数取特殊值时,这种AT-β-Spline曲线曲面可以精确地表示圆锥曲线曲面.从实验的结果来看,本文构造的AT-β-Spline曲线曲面是实用的有效的.

THE CONSTRUCTION OF AT-β-SPLINE CURVE AND SURFACE MEETING WITH GEOMETRIC CONTINUITY

In order to modify the given curves and surfaces, the algebraic trigonometric polynomial spline curves and surfaces with two kinds of shape parameters are constructed and satisfy geometric continuity. It referred to as AT-β-Spline. The AT-β-Spline not only inherit the properties of the ordinary trigonometric spline curves, but also has global and local adjustable. When two kinds of shape parameters are given in a given range, an AT-β-Spline curve satisfies first order geometric continuity. In particular when two kinds of shape parameters-1 ≤ α ≤ 1, μ_i=λ_i+1 or μ_i=λ_i=μ_i+1=λ_i+1=λ, The AT-β-Spline curves with parameters meet the C¹ ∩ G² continuity. By using the singular blending theory, we had constructed the interpolation AT-β-Spline curve which meet with the C¹ ∩ G² continuity, the curve can solve the geometric continuity of the curve and inverse algorithm of the curve. At the same time, the paper have constructed the rotation surface and discussed the influence of shape parameters. This curve and surface can be represented by an example of a conic curve or surface. From the experimental results, the curve and surface constructed by this method is practical and effective.