I noticed that somehow I did not finish with the case So today, without further ado, I am posting the algorithm.
We have the body with and we are considering the case with , where is, as always, the ratio of the doubled kinetic energy to angular momentum squared.
Then we define
I use the above formulas to draw a stereographic projection of one particular path. So, I take , and do the parametric plot of the curve in
I show below two plots. One with and one with For this selected value of the time between consecutive flips, given by the formula
So, for we have somewhat less than 200 flips, and the lines are getting rather densely packed in certain regions.
Notice that is just one geodesic line, geometrically speaking the straightest possible line in the geometry determined by the inertial properties of the body.
It is this “geometry”that will become the main subject of the future notes.
How can such a line be “straight”? Well, it is….