[math-fun] QUESTION: Rolling a ball along a closed curve

Suppose we roll a unit-radius ball without slipping along a closed curve C on the xy-plane in 3-space. This will have the net effect of applying some rotation to the ball. For instance, if C is an equilateral triangle of side-length = π, the net rotation will switch the N and S poles of the ball. QUESTION: What is the shortest closed curve C shorter than 3π that also switches the ball's poles? (Or at least, what is the inf of the lengths of all closed curves C that switch the poles?) --Dan