The asteroid surveying problem and other puzzles
Timothy M. Chan, Alexander Golynski, Alejandro López-Ortíz, Claude-Guy Quimper · 2003
We consider two variants of the well-known "sailor in the fog" puzzle. The first version (the "asteroid surveying" problem) is set in three dimensions and asks for the shortest curve that starts at the origin and intersects all planes at unit distance from the origin. Several possible solutions are suggested in the video, including a curve of length less than 12.08. The second version (the "river shore" problem) asks for the shortest curve in the plane that has unit width. A solution of length 2.2782 is described, which we have proved to be optimal.