Integrability for the Derivative Formulas of Rotation Minimizing Flame in Euclidean 3-S pace and Its Applications

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Yerlikaya F., Aydemir İ.

INTERNATIONAL ELECTRONIC JOURNAL OF GEOMETRY, vol.13, no.1, pp.116-128, 2020 (ESCI) identifier

  • 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), TR DİZİN (ULAKBİM)
  • Page Numbers: pp.116-128
  • Keywords: Bishop slant helix, Darboux helix, rotation minimizing frame, Euclidean 3-space, POSITION VECTORS, CURVES, HELICES
  • 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.