I do have a problem understanding the concept of geodesics. As I understand it a geodesic is the shortest distance between two points on any manifolds. Let's consider a spherical earth as depicted in this figure from the Wikipedia. As I read in the Wikipedia article about geodesics they are always part of a great circle. Though, if I am not mistaken, the shortest distance between two points on the same latitude (dashed lines in the figure) is to follow this latitude, is it not? But latitudes are not great circles (which are the solid lines in the figure).
So if following the latitude is the shortest path between two points on the same latitude, i.e. the geodesic, this would not be part of a great circle, since latitudes are not great circles which contradicts that geodesics are always part of a great circle. Can someone please point out my error or help me to understand the terminology better?
Thank you very much.
Best Answer
Lines of latitude are not shortest paths, except for the Equator. Here is a counterexample that's easy to visualize.
Suppose you're very close to the North Pole, say at $89^\circ$ North, and you want to go to the opposite end of the same latitude. At this scale, the line of latitude looks like a small circle. It's shorter to cut straight across the diameter, passing over the North Pole, than to walk halfway along the circle's circumference.
This is one reason why long-distance flight paths look funny when you see them on a map. Most maps make lines of latitude be straight lines, so the actual shortest paths look bent.