Characterizations of $f$-Osculating Curves in $3$ and $4$ Dimensional Euclidean Spaces

In this paper, we introduce and study the concept of $f$-osculating curves in both three and four dimensional Euclidean spaces ($\mathbb{E}^{3}$ and $\mathbb{E}^{4}$). These curves are characterized by the condition that their $f$-position vectors lie in the osculating plane. We establish necessary and sufficient conditions for a unit-speed curve in $\mathbb{E}^3$ and $\mathbb{E}^4$ to be an $f$-osculating curve and derive relations among their curvature functions. We also examine special cases in which one or more curvature functions are constant, analyzing the behaviour of the remaining curvature functions.