Since the only geodesics that are closed on an ellipsoid are the meridian and equator, doesn't they mean that all points are (eventually) connected by a geodesi

Since the only geodesics that are closed on an ellipsoid are the meridian and equator, doesn't they mean that all points are (eventually) connected by a geodesics? Obviously this isn't what anybody means by a straight line though. I'm surprised I can't find a "short" (no longer than the largest distance between points) geodesic fitter anywhere. Sounds like a fun hour-long project using GeographicLib.