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

Abstract

Let $γ: I \rightarrow \mathbb R^n$ be a parametric curve of class $C^{n+1}$, regular of order $n$. The Frenet-Serret apparatus of $γ$ at $γ(t)$ consists of a frame $e_1(t), \dots , e_n(t)$ and generalized curvature values $κ_1(t), \dots, κ_{n-1}(t)$. Associated with each point of $γ$ there are also local singular vectors $u_1(t), \dots, u_n(t)$ and local singular values $σ_1(t), \dots, σ_{n}(t)$. This local information is obtained by considering a limit, as $ε$ goes to zero, of covariance matrices defined along $γ$ within an $ε$-ball centered at $γ(t)$. We prove that for each $t\in I$, the Frenet-Serret frame and the local singular vectors agree at $γ(t)$ 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 $γ(t)\subset \mathbb R^n$ for any $n\in\mathbb N$ then, for each $i$ between $2$ and $n$, $κ_{i-1}(t)=\sqrt{a_{i-1}}\frac{σ_{i}(t)}{σ_1(t) σ_{i-1}(t)}$ 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.