Natural and conjugate mates of Frenet curves in three-dimensional Lie group

Year 2021, Volume: 70 Issue: 1, 522 - 540, 30.06.2021

Mahmut Mak

https://doi.org/10.31801/cfsuasmas.785489

Cited By: 3

Abstract

In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.

Keywords

Natural mate, conjugate mate, helix, slant helix, spherical curve, rectifying curve, Salkowski curve, anti-Salkowski curve

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