论文标题
$ \ mathbb {r}^n $的几何赫米特插值通过改进
Geometric Hermite Interpolation in $\mathbb{R}^n$ by Refinements
论文作者
论文摘要
我们描述了一种基于几何赫米特数据的近似高维曲线的广泛的操作员的一般方法。几何图案数据由点样本及其单位长度的切线向量组成。扩展了函数的经典Hermite插值，这种几何赫米特问题近年来已经流行，并在2D平面和3D空间中点燃了一系列解决方案。在这里，我们提出了一种近似曲线的方法，该方法在任何维度上都是有效的。我们方法的基本构建基础是Hermite的平均值 - 本文中引入的概念。我们通过说明性插值细分方案提供了这样一个平均值的示例，并显示了该细分方案的限制如何继承平均值的几何特性。最后，我们证明了该细分方案的收敛性，该方案的极限将几何图形数据插值并近似采样曲线。我们以各种数值示例结束了本文，阐明了我们方法的优势。
We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of unit length. Extending the classical Hermite interpolation of functions, this geometric Hermite problem has become popular in recent years and has ignited a series of solutions in the 2D plane and 3D space. Here, we present a method for approximating curves, which is valid in any dimension. A basic building block of our approach is a Hermite average - a notion introduced in this paper. We provide an example of such an average and show, via an illustrative interpolating subdivision scheme, how the limits of the subdivision scheme inherit geometric properties of the average. Finally, we prove the convergence of this subdivision scheme, whose limit interpolates the geometric Hermite data and approximates the sampled curve. We conclude the paper with various numerical examples that elucidate the advantages of our approach.
