Torus equations?

[Guv]

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).

[RAEsquivelC]

Guv posted:

"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)."
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Seems to me that the value of the shortest distance is one, but, the paths are two: one that crosses the inner equator, and one that sort of "reflects back" just as it reaches the inner equator, without crossing it.

Later: Hmm, thinking more about it, I guess one should say there are eight paths, all the same length.

[Guv]

RAEsquivelC: I did enough analysis of geometry on a circular torus to determine that there are the following types of geodesics. The inner and outer equators are geodesics. For any two points on the inner equator, an arc of the inner equator is the shortest distance. For many, but not all, pairs of points on the outer equator, an arc of the outer equator is the shortest distance. Polar circles are geodesics. An arc of a polar circle is the shortest distance between two points with the same longitude. Between any two distinct points on the outer equator, there are infinitely many helix-like geodesics, which cross the inner equator, some circling many times, crossing both the inner and outer equator more than once. Obviously, the one which crosses the inner equator only once is the shortest of these. As the points approach each other, the helix-like geodesic approaches a polar circle. Between certain points on the outer equator, there are sine-wave-like geodesics, which move toward the inner equator (never crossing it) and turn back toward the outer equator, crossing it and moving toward the inner equator in the other direction, only to return before crossing the inner equator. These geodesics are periodic like a sine wave. There is a geodesic which is a compromise between the sine-wave-like and the helix-like geodesics. I refer to it as the undecided geodesic, not being able to choose between being helix-like or sine-wave-like. It starts at a point on the outer equator and heads toward the inner equator. This geodesic is asymptotic to the inner equator. It is skew-symmetric to the outer equator, there being a segment which starts toward the north polar circle, and an oppositely directed segment which starts toward the south polar circle. The two segments continue indefinitely, getting closer and closer to the inner equator, but never meeting there. The two segments seem to circle the inner equator in opposite directions, and become indistinguishable from the inner equator. All shortest distance curves and all geodesics between two points on the torus are arcs of one of the above curves.Other odds and ends of data about torus geometry.There is a helix-like geodesic which starts at a point on the outer equator and returns to that point after traveling 360 degrees (2Pi radians) of latitude and 360 degrees of longitude, crossing the inner equator once. I assume that this geodesic crosses the inner equator at longitude 180 degrees (Pi radians).

There are geodesics belonging to the helix-like family which do not look anything like a helix. These approach the inner equator, seem to circle it many times and then cross it. After crossing, they seem to circle the inner equator many times before heading toward the outer equator. Many return to the starting point on the outer equator and retrace their initial path if continued. Many never return to the same point on the outer equator. These geodesics seem more like the undecided geodesic than a helix-like geodesic.

Similarly there are geodesics belonging to the sine-wave-like family which seem to circle the inner equator many times before returning to the outer equator. These seem more like the undecided geodesic than a sine-wave-like geodesic. There are many of these which return to the starting point on the outer equator and retrace the initial path if continued. There are many which never return to the same point on the outer equator.

There is a helix-like geodesic which travels 360 degrees of latitude while traveling 180 degrees of longitude. There is a sine-wave-like geodesic which returns to the outer equator 180 degrees from the starting point on the outer equator. One of these two geodesics is the shortest distance between points on the outer equator 180 degrees apart (half way around). I do not think that they are equal in length, but they could be.

There is a minimum period for the sine-wave-like geodesics. For two nearby points on the outer equator, an arc of the outer equator is the shortest distance. For other pairs of points on the outer equator, a sine-wave-like geodesic is the shortest distance. There is some distance for which the outer equator distance and the sine-wave-like distance are equal. This distance is the minimum period possible for a sine-wave-like geodesic. On the torus (as is true for various other curved surfaces), there can be two different curves paths equal in length and both the shortest distance between two points.