In plane for a smooth non self intersecting curve $C$ the tubular neighbourhood can be constructed with non intersecting starigh line segments normal to the curve from points of the curve. If we use the fact that the $\varepsilon$-tubular neighbourhood is constructed diffeomorphically by extending the normals we could arrive at a necessary condition that $\varepsilon < 1/K$, where $K=\sup_{p\in C} k(p)$ and $k(p)$ is the curvature of curve at $p$.

Suppose the curve $C$ is in a surface embedded in $R^3$. The tubular neighbourhood can be constructed similarly along the normal geodesics to the curve. Is it true that for $K=\sup k_g(p)$ where $k_g(p)$ is geodesic curvature of curve at $p$, the necessary condition is $\varepsilon < 1/K$?