I'm trying to find the geodesics on an ellipsoid and I have a few questions..
First to begin I am trying to understand the different types of ellipsoids. I would have liked to approach finding the geodesics using Clairaut's relation so I'm wondering if all ellipsoids are surfaces of revolution.
I know there are cases: for an ellipsoid centered at the origin with equation $\frac{x^2}{a^2} +\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1$ and parametric equation $\textbf{x}(u,v) = (a\cos(u)\sin(v),b\sin(u)\sin(v),c\cos(v))$ where $u \in [0,2\pi), v \in [0,\pi)$ so as a surface of revolution I'm taking my equation to be $f(v)= (0, \sin(v),\cos(v))$ $v \in [0,\pi)$ and then rotating it.
1) the sphere which I'm not so concerned about since I know it's geodesics are great circles.
2) The case where a = b and c is different
3) The case where a,b < c
4) The case where a,b > c
So I'm just wondering can all these cases be generalized to surfaces of revolution...
From there I am going to solve the geodesic equations which I know is a problem in it of itself, but really I just want to know is looking at the geodesics in these 4 cases is a good way to start and also if the way I'm looking at the ellipsoid as a surface of revolution is correct... (I'm guessing in the last two cases is where I have an issue)