The weak Frenet frame of non-smooth curves with finite total curvature and absolute torsion
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The weak Frenet frame of non‑smooth curves with finite total curvature and absolute torsion Domenico Mucci1 · Alberto Saracco1 Received: 17 February 2019 / Accepted: 15 March 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Binormal · Total absolute torsion · Polygonals · Non-smooth curves Mathematics Subject Classification 53A04
1 Introduction In classical differential geometry, it sometimes happens that the geometry of a proof can become obscured by analysis. This statement by Penna [11], which may be referred, e.g., to the classical proof of the Gauss-Bonnet theorem, suggests to apply piecewise linear methods in order to make the geometry of a proof completely transparent. For this purpose, by using the geometric description of the torsion of a smooth curve, Penna [11] gave in 1980 a suitable definition of torsion for a polygonal curve of the * Alberto Saracco [email protected] Domenico Mucci [email protected] 1
Dipartimento di Scienze Matematiche, Fisiche ed Informatiche, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy
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Euclidean space ℝ3 , and used piecewise linear methods and homotopy arguments to produce an illustrative proof of the well-known property that the total torsion of any closed unit speed regular curve of the unit sphere 𝕊2 is equal to zero. Differently to the smooth case, the polygonal torsion is a function of the segments. His definition, in fact, relies on the notion of binormal vector at the interior vertices. Since the angle between consecutive discrete binormals describes the movements of the “discrete osculating planes” of the polygonal, binormal vectors naturally live in the projective plane ℝℙ2 , see Sect. 2. We recall here that Milnor [8] defined the tangent indicatrix, or tantrix, of a polygonal P as the geodesic polygonal 𝔱P of the Gauss sphere 𝕊2 obtained by connecting with oriented geodesic arcs t
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