  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.
