In this paper, we present a systematic study on the characterization of osculating curves under isometries using the Darboux frame in place of the Frenet frame. We further investigate the conditions for the invariance of such curves on regular surfaces embedded in Euclidean 3-space under isometric transformations. The present analysis specifically focuses on the application of the Darboux frame {𝒯, 𝒫, 𝒰}. Furthermore, we investigate how the position vector of an osculating curve on regular surfaces is affected by isometric transformations, analyzing its behavior under different orientations, specifically along the unit tangent vector 𝒯, the unit normal 𝒰, and 𝒫 = 𝒰 × 𝒯.