Osculating surfaces with modified orthogonal frame in Euclidean 3-space

This work develops a class of osculating surfaces in three-dimensional Euclidean space using a modified orthogonal frame defined along curves with non-zero curvature. We begin by deriving the connections between the classical Frenet frame and the modified orthogonal frame adapted to curvature. Within this setting, the fundamental geometric quantities of the resulting osculating surfaces, including their singular behavior, Gaussian curvature, mean curvature, and fundamental forms, are explicitly determined. Moreover, criteria are established for when the parameter curves of the associated osculating surfaces represent geodesics, asymptotic curves, or curvature lines. Finally, to demonstrate the applicability of the theoretical results, a representative example of an osculating surface constructed using both the Frenet frame and the modified orthogonal frame is provided.