a closed geodesic on the torus

solution of a second order coupled system of ordinary differential equations for the position coordinates on the surface
of a curve which locally minimizes the length of the curve segment between successive points on it
(called a geodesic)

The curve starts at the intersection of the blue and red circles aimed upward slightly to the right of the blue circle. It winds around the inner hole of the torus just above the inner equator (green circle) one loop and then comes back over the top and then slips underneath the bottom side where it repeats the mirror image of the top behavior (one loop around the hole under the inner equator) and finally returns exactly to its starting point (so that it is a closed curve). This is a very special curve, since usually such curves never return to their starting point.

Indeed it makes 1 complete oscillation about the outer equator while making 3 revolutions around the vertical axis through the central hole of the torus, but not passing through the inner equator (it only "skims" the inner equator!), making it a [1,3;0] class closed geodesic. A closed geodesic can be characterized by the triplet [m,n;p] making n revolutions about the vertical axis, while making m oscillations away from the outer equator, either passing through the inner equator (p = 1) or not (p = 0), letting [0,1;1] describe the exceptional inner equator.

For more details than you want to see, go to the torus page.