This question is from Introduction to Classical Mechanics, David Morin.
A mountain climber wishes to climb up a frictionless conical mountain. He wants to do this by throwing a lasso (a rope with a loop) over the top and climbing up along the rope. Assume that the mountain climber is of negligible height, so that the rope lies along the mountain, as shown.
At the bottom of the mountain are two stores. One sells “cheap” lassos (made of a segment of rope tied to loop of rope of fixed length). The other sells “deluxe” lassos (made of one piece of rope with a loop of variable length; the loop’s length may change without any friction of the rope with itself). When viewed from the side, the conical mountain has an angle α at its peak. For what angles α can the climber climb up along the mountain if he uses a “cheap” lasso?A“deluxe” lasso? (Hint: The answer in the “cheap” case isn’t α < 90◦.)
In the fourth paragraph, I couldn't understand why the rope should take the shortest distance between two points when the cone is on the plane. Why no friction implies that?