On Characterizations of Osculating Curves using Darboux Frame in Minkowski 3-Space

Year 2026, Volume: 14 Issue: 1 , 242 - 249 , 30.04.2026

Kemal Eren , Mahmutcan Carlı , Soley Ersoy

https://izlik.org/JA27AT22DC

Abstract

In this paper, we investigate osculating curves in Minkowski 3-space by means of the Darboux frame associated with a non-null curve lying on a surface. Moreover, we introduce and construct two distinct classes of osculating curves, namely type-1 and type-2 osculating curves. Using the Darboux frame, we derive necessary and sufficient conditions under which a non-null curve lying on a surface becomes an osculating curve, expressed in terms of the geodesic curvature ${k_g}$, normal curvature ${k_n}$, and geodesic torsion ${\tau _g}$. As a consequence of these conditions, several corollaries and theorems concerning type-1 and type-2 osculating curves are established. Finally, illustrative examples supporting the theoretical results are presented, and graphical visualizations of the obtained curves are provided to demonstrate their geometric behavior.

Keywords

Darboux frame , osculating curve , geodesic torsion , geodesic curvature , normal curvature

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