# Do geodesics close on an ellipsoid?

I want to calculate the shortest distance between two points (Point A and Point B) on the ellipsoid surface. For this I need to use geodetic passing through these two points. Well, if I continue this geodesic that I defined between 2 points along the ellipsoid, will the geodetic converge at point A again?

In some sources it was said that only meridians and equator are closed geodesics, this confused me.

• en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid has lots of good info and plenty of diagrams. FWIW, a major contributor to the Wikipedia ellipsoid articles is Dr C. F. F. Karney, author of the excellent free geographiclib. Jun 7, 2021 at 13:24
• @PM2Ring this appears to be a python wrapper for it pypi.org/project/geographiclib Cool!
– uhoh
Jun 8, 2021 at 0:26
• @uhoh Indeed! I used geographiclib in Python here: math.stackexchange.com/a/1340899/207316 See the end of that answer for a GitHub link. Jun 8, 2021 at 9:05
• @PM2Ring as my copy of Smart's Spherical Astrometry is currently on the other side of the Earth, knowing that will come in very handy, in fact I can now use python to calculate just how far it is away from be including Earth's oblateness :-) Speaking of Math SE I've been having some fun recently 1 and now 2
– uhoh
Jun 8, 2021 at 9:35

No, in general, they don't close. (Though, as you say, some geodesics do.) Consider, for example, an oblate ellipsoid of revolution, and take two points on its equator such that the angle $$\alpha$$ between them is not a rational multiple of $$\pi$$. On a sphere, the only geodesic passing through tho points on the equator is the equator; but in our case, since the ellipsoid is oblate, if $$\alpha$$ is large enough (i.e., close enough to $$\pi$$), then there are shorter paths connecting the two points, so there are other geodesics passing through the two points. If we take one of these geodesics — let's say the one going through the northern hemisphere — and continue it past one of the two points into the southern hemisphere, it will behave symmetrically there and intersect the equator after another angle $$\alpha$$. And then after another $$\alpha$$, and so on. Since $$\alpha$$ is not a rational multiple of $$\pi$$, there will be infinitely many points where this geodesic intesects the equator, so it will never close.