Orthogonal polynomials on planar cubic curves

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

This paper presents a nonconforming finite element scheme for the planar biharmonic equation which applis piecewise cubic polynomials ($P_3$) and possesses $\mathcal{O}(h^2)$ convergence rate in energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the $H^4$ functions with $\mathcal{O}(h^2)$ accuracy in broken $H^2$ norm. Besides, an equivalence $(\nabla_h^2\ \cdot,\nabla_h^2\ \cdot)=(Δ_h\ \cdot,Δ_h\ \cdot)$, which is usually not true for nonc…

Orthogonal polynomials on a class of planar algebraic curves

We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form $y^m = ϕ(x)$ in $\mathbb{R}^2$ where $m = 1, 2$ and $ϕ$ is a polynomial of arbitrary degree $d$, in terms of univariate semiclassical OPs. We compute connection coeffeicients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree $0, \ldots, N$ are computed via the Lanczos algorithm in $O(Nd^4)$ operations.

平面有理三次贝塞尔曲线隐式表示的一个基底

A basis for the implicit representation of planar rational cubic Bézier curves

本文提出一种通过定义新基底来求解平面有理参数三次曲线隐式方程的方法，该基底由四个三次双变量多项式组成，并基于曲线的贝塞尔控制点。同时给出了隐式曲线系数的显式公式，并讨论了曲线几何行为的描述。

平面 $C^1$ 三次样条顶点插值的一个反例

A counterexample to vertex interpolation by planar $C^1$ cubic splines

构造一个41顶点56三角形的三角剖分，证明其上三次C^1样条无法插值所有顶点数据，通过显式线性关系否证Alfeld猜想，无需矩阵秩计算。