The weak Frenet frame of non-smooth curves with finite total curvature and absolute torsion
| Autor: | Alberto Saracco, Domenico Mucci |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Mathematics - Differential Geometry
Euclidean space Applied Mathematics Frenet–Serret formulas 010102 general mathematics Mathematical analysis Torsion (mechanics) Tangent Curvature 01 natural sciences Measure (mathematics) 53A04 Differential Geometry (math.DG) 0103 physical sciences FOS: Mathematics Total curvature 010307 mathematical physics Projective plane 0101 mathematics Mathematics |
| Zdroj: | Annali di Matematica Pura ed Applicata (1923 -). 199:2459-2488 |
| ISSN: | 1618-1891 0373-3114 |
| DOI: | 10.1007/s10231-020-00976-5 |
| Popis: | 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 notation 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 with positive curvature, the weak binormal and normal yield (up to a lifting) the classical notions of binormal and normal. Comment: 18 pages, 2 figures |
| Databáze: | OpenAIRE |
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