Years ago I was interested in the gravitational field around a torus, and the geodesics on its surface.

I was never able to develop an analytical equation for the gravitational field, and do not think one can be developed. If anyone knows of such I would be interested. A numerical solution using integrals does not interest me.

I did develop a usable differential equation for geodesics, and would like to avoid doing it over again. Does anybody have such an equation handy?

By the way: Consider the shortest distance curve from a point on the "outer equator" of the torus to a point 180 degrees away on the outer equator. Does this curve cross the "inner equator" or go toward it and then return to the outer equator? I know it must do one or the other, but never worked out which when I had the equations (I was using a hand calculator and it would have taken too long).