Position Vectors of Curves With Recpect to Darboux Frame in the Galilean Space $G_3$

TL;DR

This paper derives the position vectors of curves on surfaces in Galilean space relative to the Darboux frame, expressing them through curvature and torsion, and explores special curves with illustrative examples.

Contribution

It provides explicit formulas for position vectors of curves in Galilean space with respect to Darboux frame, including special curves like geodesics and asymptotic lines.

Findings

Position vectors expressed in terms of geodesic, normal curvature, and torsion.

Definitions of position vectors for geodesic, asymptotic, and normal lines.

Examples with graphical representations of the curves.

Abstract

In this paper, we investigate the position vector of a curve on the surface in the Galilean 3-space G^3. Firstly, the position vector of a curve with respect to the Darboux frame is determined. Secondly, we obtain the standard representation of the position vector of the curve with respect to Darboux frame in terms of the geodesic, normal curvature and geodesic torsion. As a result of this, we define the position vectors of geodesic, asymptotic and normal line along with some special curves with respect to Darboux frame. Finally, we elaborate on some examples and provide their graphs.