Geometry of Curves in $\mathbb R^n$, Singular Value Decomposition, and Hankel Determinants

Abstract

Let $\gamma: I \rightarrow \mathbb R^n$ be a parametric curve of class $C^{n+1}$, regular of order $n$. The Frenet-Serret apparatus of $\gamma$ at $\gamma(t)$ consists of a frame $e_1(t), \dots , e_n(t)$ and generalized curvature values $\kappa_1(t), \dots, \kappa_{n-1}(t)$. Associated with each point of $\gamma$ there are also local singular vectors $u_1(t), \dots, u_n(t)$ and local singular values $\sigma_1(t), \dots, \sigma_{n}(t)$. 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 $t\in I$, 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 $t$. More precisely, we show that if $\gamma(t)\subset \mathbb R^n$ for any $n\in\mathbb N$ then, for each $i$ between $2$ and $n$, $\kappa_{i-1}(t)=\sqrt{a_{i-1}}\frac{\sigma_{i}(t)}{\sigma_1(t) \sigma_{i-1}(t)}$ with $a_{i-1} = \left(\frac{i}{i+(-1)^i}\right)^2 {\frac{4i^2-1}{3}}$. For this 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.