Shortest inspection curves for a sphere

What is the form of the shortest curve $C$ going outside the unit sphere $S$ in $\mathbb R^3$ such that passing along $C$ we can see all points of $S$ from outside? How will the form of $C$ change if we require that $C$ have one of its (or both) endpoints on $S$? A solution to the latter problem also answers the following question. You are in a half-space at a unit distance from the boundary plane $P$, but do not know where $P$ is. What is the shortest space curve $C$ such that going along $C$ you certainly will come to $P$? Geometric arguments are given suggesting that the required curves should be looked for in certain classes depending on several parameters. A computer analysis yields the best curves in the classes. Some other questions are solved in a similar way.