Abstract
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.
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Acknowledgements
D.M. wishes to thank his seventeen-year-old son, Giovanni Mucci, for his useful help in pointing out to us some interesting phenomena concerning spherical geometry that he learned by himself. We also thank the referee for his/her suggestions that allowed us to improve the first version of the paper and to obtain the results contained in the last section. The research of D.M. was partially supported by the GNAMPA of INDAM. The research of A.S. was partially supported by the GNSAGA of INDAM.
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Mucci, D., Saracco, A. The weak Frenet frame of non-smooth curves with finite total curvature and absolute torsion. Annali di Matematica 199, 2459–2488 (2020). https://doi.org/10.1007/s10231-020-00976-5
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DOI: https://doi.org/10.1007/s10231-020-00976-5