Frenet-Serret and the Estimation of Curvature and Torsion

Abstract

In this paper we approach the problem of analyzing space-time curves. In terms of classical geometry, the characterization of space-curves can be summarized in terms of a differential equation involving functional parameters curvature and torsion whose origins are from the Frenet-Serret framework. In particular, curvature measures the rate of change of the angle which nearby tangents make with the tangent at some point. In the situation of a straight line, curvature is zero. Torsion measures the twisting of a curve, and the vanishing of torsion describes a curve whose three dimensional range is restricted to a two-dimensional plane. By using splines, we provide consistent estimators of curves and in turn, this provides consistent estimators of curvature and torsion. We illustrate the usefulness of this approach on a biomechanics application.