Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications

Year 2020, Volume: 13 Issue: 1, 116 - 128, 30.01.2020

Fırat Yerlikaya , İsmail Aydemir

https://doi.org/10.36890/iejg.621588

Cited By: 1

Abstract

We analyze integrability for the derivative formulas of the rotation minimizing frame in the Euclidean 3-space from a viewpoint of rotations around axes of the natural coordinate system. We give a theorem that presents only one component of the indirect solution of the rotation minimizing formulas. Using this theorem, we find a lemma which states the necessary condition for the indirect solution to be a steady solution. As an application of the lemma, the natural representation of the position vector field of a smooth curve whose the rotation minimizing vector field (or the Darboux vector field) makes a constant angle with a fixed straight line in space is obtained. Also, we realize that general helices using the position vector field consist of slant helices and Darboux helices in the sense of Bishop.

Keywords

Bishop slant helix, Darboux helix, rotation minimizing frame, Euclidean 3-space

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