Integrability for the Derivative Formulas of Rotation Minimizing Flame in Euclidean 3-S pace and Its Applications
INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY, vol.13, no.1, pp.116-128, 2020 (ESCI, Scopus, TRDizin)
- Publication Type: Article / Article
- Volume: 13 Issue: 1
- Publication Date: 2020
- Doi Number: 10.36890/iejg.621588
- Journal Name: INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY
- Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, TR DİZİN (ULAKBİM)
- Page Numbers: pp.116-128
- Keywords: Bishop slant helix, Darboux helix, rotation minimizing frame, Euclidean 3-space, POSITION VECTORS, CURVES, HELICES
- Open Archive Collection: AVESIS Open Access Collection
- Ondokuz Mayıs University Affiliated: Yes
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.