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

Abstract

Let $\gamma :I\to {\mathbb{R}}^n$ 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.

References in corpus (1)

• Degrees of projections of rank loci