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 y:I--rn be a parametric curve of class Cn+1, regular of order n. The Frenet-Serret apparatus of y at y(t) consists of a frame e1(t),...,en(t) and generalized curvature values kn(t),..., kn-1(t). Associated with each point of y there are also local singular vectors u1(t),..., un(t)
and local singular values
. This local information is obtained by considering a limit, as ? goes to zero, of covariance matrices defined along ? within an ?-ball centered at
. We prove that for each
, the Frenet-Serret frame and the local singular vectors agree at
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.
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