Calculus of Variations and Geometric Measure Theory

D. Mucci - A. Saracco

The weak Frenet frame of non-smooth curves with finite total curvature and absolute torsion

created by mucci on 10 Jan 2019
modified on 17 May 2020


Published Paper

Inserted: 10 jan 2019
Last Updated: 17 may 2020

Journal: Annali di Matematica Pura ed Applicata
Year: 2020

Published online: 07 April 2020


We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notion for polygonal curves, where the polarity property is exploited, and then make use of a density argument. Both our weak binormal and normal are rectifiable curves which naturally live in the projective plane. In particular, the length of the weak binormal agrees with the total absolute torsion of the given curve. Moreover, the weak normal is the vector product of suitable parameterizations of the tangent indicatrix and of the weak binormal. In the case of smooth curves, the weak binormal and normal yield (up to a lifting) the classical notions of binormal and normal. Finally, the torsion force is introduced: similarly as for the curvature force, it is a finite measure obtained by performing the tangential variation of the length of the tangent indicatrix in the Gauss sphere.

Keywords: binormal, total absolute torsion, polygonals, non-smooth curves