基于参数速度逼近的等距曲线有理逼近

Rational Approximation of Offset Curves by Parametric Speed Approximation

该文提出了曲线的参数速度逼近问题 ,指出等距曲线逼近的关键在于参数速度的逼近 ,并用两种方式来实现它 .首先 ,以法矢方向曲线的控制顶点模长为 Bézier纵标构造 Bernstein多项式 ,以它来逼近曲线的参数速度 ,给出了相应的几何方式的等距逼近算法 ,进一步利用法矢方向曲线的升阶获得了高精度逼近 .其次 ,基于参数速度的 L egendre多项式逼近和插值区间端点的 Jacobi多项式逼近 ,导出了保持法矢平移方向的两种代数方式的等距有理逼近算法 .

The problem of parametric speed approximation of a curve is raised in this paper. The authors point out that the crux of offset curve approximation lies in the approximation of parametric speed, and two methods are provided. The parametric speed of the curve is firstly approximated by the Bernstein polynomial, which takes the lengths of control point vectors of the direction curve of normal as Bézier coordinates. Then the corresponding geometric offset approximation algorithm is given. Moreover, an offset approximation with high precision is obtained by degree elevation of the direction curve of normal. Based on the Legendre polynomials approximation and Jacobi polynomials approximation with endpoints interpolation of parametric speed of the curve, two algebraic rational approximation algorithms of offset curves, which preserve the direction of normal, are derived.