On optimal polynomial geometric interpolation of circular arcs according to the Hausdorff distance

基于曲率的圆弧最优多项式几何插值

Curvature-Based Optimal Polynomial Geometric Interpolation of Circular Arcs

本文针对圆弧的低次参数多项式逼近，提出以曲率误差为最优准则并在边界满足几何连续性的插值方法，分析了抛物 $G^0$、三次 $G^1$ 和四次 $G^2$ 情形，并与径向误差准则进行比较。

Arc length preserving approximation of circular arcs by Pythagorean-hodograph curves of degree seven

In this paper interpolation of two planar points, corresponding tangent directions and curvatures with Pythagorean-hodograph (PH) curves of degree seven preserving an arc length is considered. A general approach using complex representation of PH curves is presented and a detailed analysis of the problem for data arising from a circular arc is provided. In the case of several solutions some criteria for the selection of the most appropriate one are described and an asymptotic analysis is given. Several numerical examples are included which confirm theoretical results.

二维多形状参数开放弧弹性散射的降基方法

Reduced Basis Method for the Elastic Scattering by Multiple Shape-Parametric Open Arcs in Two Dimensions

针对二维多开放弧弹性散射问题，提出一种基于降基方法和改进经验插值法的快速求解方法，通过离线构建单弧降阶基和在线组合，实现高精度与高计算效率。

Order of uniform approximation by polynomial interpolation in the complex plane and beyond

For Lagrange polynomial interpolation on open arcs $X=γ$ in $\mathbb{C}$, it is well-known that the Lebesgue constant for the family of Chebyshev points ${\bf{x}}_n:=\{x_{n,j}\}^{n}_{j=0}$ on $[-1,1]\subset \mathbb{R}$ has growth order of $O(log(n))$. The same growth order was shown in [45] for the Lebesgue constant of the family ${\bf {z^{**}_n}}:=\{z_{n,j}^{**}\}^{n}_{j=0}$ of some properly adjusted Fejér points on a rectifiable smooth open arc $γ\subset \mathbb{C}$. On the other hand, in our recent work [15], it was observed that if the smooth open arc $γ$ is replaced by an $L$-shape arc $γ…