Let $M$ be a compact connected surface (possibly with boundary) in $\mathbb{R}^3$ and $\epsilon>0$ a constant.

Is there (and if there's not, what conditions on ($M$, $\epsilon$) should we add to have such existence) a curve $\gamma$ on $M$ with minimal length such that for any point $x\in M$, there is a point $\gamma(t)$ on the curve such that $d(\gamma(t),x)\leq \epsilon$.

I am also interested, in the case of positive results of existence, on the method to obtain explicitly such minimal length curves.

Let $M$ be a compact connected (smooth) surface (possibly with boundary) in $\mathbb{R}^3$ and $\epsilon>0$ a constant.

Is there (and if there's not, what conditions on ($M$, $\epsilon$) should we add to have such existence) a (smooth) curve $\gamma$ on $M$ with minimal length such that for any point $x\in M$, there is a point $\gamma(t)$ on the curve such that $d(\gamma(t),x)\leq \epsilon$.

I am also interested, in the case of positive results of existence, on the method to obtain explicitly such minimal length curves.