Geometry of curves in Rn from the local singular value decomposition
Geometry of curves in Rn from the local singular value decomposition
复制标题
DOI：
10.1016/j.laa.2019.02.006
复制
发表时间：
2019-06-15
影响因子：
1.1
通讯作者：
Draper, Bruce
中科院分区：
文献类型：
--
作者：
Alvarez-Vizoso, J.;Arn, Robert;Draper, Bruce
我们建立了局部奇异值分解与n维曲线几何之间的联系。特别地，我们将左奇异向量与Frenet-Serret标架联系起来，并将广义曲率与奇异值联系起来。具体地说，设Gamma：i-gt；R-n是一条n阶正则的Cn+1类参数曲线。Gamma(T)的Frenet-Serret装置由一个框架e(1)(T)，…，e(N)(T)和广义曲率值kappa(1)(T)，…，kappa(n-1)(T)组成。与Gamma的每个点相关联的还有局部奇异向量u(1)(T)，…，u(N)(T)和局部奇异值sigma(1)(T)，…，sigma(N)(T)。该局部信息是通过考虑在以伽马(T)为中心的埃球内沿着伽马定义的协方差矩阵的极限来获得的。证明了对于任意t是i的元素，Frenet-Serret标架和局部奇异向量在Gamma(T)处一致，并且曲率函数在t处的值可以表示为在t处局部奇异值之比的固定倍数.为了建立这一结果，我们利用一次正交多项式理论和矩序列理论证明了一类Hankel行列式序列的递推关系的一般公式.(C)2019年由Elsevier Inc.出版。
We establish a connection between the local singular value decomposition and the geometry of n-dimensional curves. In particular, we link the left singular vectors to the Frenet-Serret frame, and the generalized curvatures to the singular values. Specifically, let gamma : I -> R-n be a parametric curve of class Cn+1, regular of order n. The Frenet-Serret apparatus of gamma at gamma(t) consists of a frame e(1)(t), ... , e(n)(t) and generalized curvature values kappa(1)(t), ... , kappa(n-1)(t). Associated with each point of gamma there are also local singular vectors u(1)(t), ... , u(n)(t) and local singular values sigma(1)(t), ... , sigma(n)(t). This local information is obtained by considering a limit, as epsilon goes to zero, of covariance matrices defined along gamma within an epsilon-ball centered at gamma(t). We prove that for each t is an element of I, the Frenet-Serret frame and the local singular vectors agree at gamma(t) and that the values of the curvature functions at t can be expressed as a fixed multiple of a ratio of local singular values at t. To establish this result we prove a general formula for the recursion relation of a certain class of sequences of Hankel determinants using the theory of monic orthogonal polynomials and moment sequences. (C) 2019 Published by Elsevier Inc.