Geometry of curves in [formula omitted] from the local singular value decomposition

Abstract

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 γ:I→Rn be a parametric curve of class Cn+1, regular of order n. The Frenet-Serret apparatus of γ at γ(t) consists of a frame e1(t),…,en(t) and generalized curvature values κ1(t),…,κn−1(t). Associated with each point of γ there are also local singular vectors u1(t),…,un(t) and local singular values σ1(t),…,σn(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∈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. 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.