Geometry of curves in Rn from the local singular value decomposition

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.