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

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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