Spinning the ball about the vertical (z) axis, and not having it move in the xy-plane, is technically “rolling without slipping”. So unless you modify your question, the length of the shortest C is zero, if I’m allowed to put N and S where I choose, or pi, if N and S start on a vertical axis. -Veit On Jul 28, 2014, at 11:26 AM, Dan Asimov <dasimov@earthlink.net> wrote:
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