Abstract
In this paper, we investigate the position vector of a curve on the surface in the Galilean 3-space G³. 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.
Highlights
The fundamental theorem of curves state that curves are determined by curvatures [1]
We investigate the position vector of a curve on the surface in the Galilean 3-space G3
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
Summary
The fundamental theorem of curves state that curves are determined by curvatures [1]. Geodesics arise from the problem of ...nding ’shortest curves’ joining two points of a surface M. It was ...rst considered by Johann Bernoulli (1697). The main aim of this study is to solve the above problem for all curves on a surface in G3 with respect to the Darboux Frame. We determine the position vector of a curve on a surface in G3 in terms of geodesic, normal curvature and geodesic torsion with respect to the Darboux and standard frame. We shall give position vectors of some special curves such as geodesic, asymptotic curve, line of curvature on a surface in G3. We want to emphasize that the results of this study can be extended to families of surfaces that have common geodesic curve
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