Around Cone Mountain puzzle

This construction is intended to help visualize the premise of a challenging puzzle, found at The track for a sightseeing train is to be constructed, making one full turn around a cone-shaped mountain. The track starts at point A on the base of the mountain and ends at point B, directly uphill from point A. Drag the various points on the 2D net to adjust the parameters of the mountain. Questions one may explore:
  • Under what conditions does the shortest track ascend first and then descend? Explain in simple language why this is sometimes shorter than a strictly-uphill track.
  • For some given set of parameters, what is the shortest possible length of the track? How long are both the ascending and descending sections of track?
  • What is the shortest track length when the central angle of the cone's net exceeds 180°?