Let $S \subset \mathbb{R}^3$ be a surface embedded in $\mathbb{R}^3$, let's say (to keep it simple) of genus zero. Let $\gamma$ be a simple, closed, oriented geodesic on $S$. Because $\gamma$ is oriented, it partitions $S$ into two "halves," $S^+$ and $S^-$. I am interested in learning properties of the curves that are equidistant from $\gamma$.

Define $$\gamma^+(\delta) = \{ x \in S^+ \;|\; d(x, \gamma) = \delta \} \;,$$ where $d(x,A)$ is the length of the shortest path on $S$ from $x$ to any point in set $A$.

Q1. Does $\gamma^+(\delta)$ have a name in the literature? Has it been studied?

Q2. Under what conditions is $\gamma^+(\delta)$ a geodesic? Presumably rather stringent conditions on $S$.

Q3. Under what conditions is $\gamma^+(\delta)$ a simple, closed curve? It might partition into several disconnected components. But perhaps for convex $S$, it is always a simple, closed curve?

Q4. Under what conditions could $\gamma^+(\delta)$ be a single point? [Revised to reflect Will Jagy's comment.] (Analogous to the north pole with respect to the equator on a (geometric) sphere.)

Thanks for pointers and help!

(Image below suggestive only!)