In plane for a smooth 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 $K=\sup_{p\in C} k(p)$. 

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 at $p$, the necessary condition is $\varepsilon < 1/K$?