Not sure if this is a matter of editorial policy, philosophy, or mathematical logic.

A year or two ago, I wrote a Visual Basic program to analyze geodesics (shortest distance curves) on the surface of a torus.

There are several types of geodesics on the surface of a torus. One type starts at the Outer Equator heads towards the North Polar Circle, reaches a maximum latitude, and returns to the Outer Equator. The numerical data from the program indicates the following.For nearby points on the Equator, the Equator itself is the shortest distance, and there is no geodesic of the type described above between the two points.


For more distance pairs of points on the Equator, a geodesic of the type described above is the shortest distance between the two points and can be significantly shorter than the Equatorial distance. For some pairs of points, there is more than one such geodesic connecting the two points, one of which is shorter than the others.



There is a critical distance along the Equator, intermediate to the above. For this distance, the Equator is the shortest distance between two points, and is equal to the distance between the points along a geodesic of the type described above. Years ago, before writing any pertinent software, I spent a lot of time thinking about geodesics on a torus. My intuition suggested the above type of geodesic, and I spent a lot of time trying to prove the critical distance property.

Perhaps in my college days, I might have come up with some proof, but theory was never my strong point. Since leaving college, I became strictly an applied mathematician and have no hope of proving anything subtle.

The numeric data from my program has convinced me of the validity of the critical distance property, but I wonder if such analysis can be considered a proof.

Well, things like this are always useful, but mathematicians would like a rigorous proof. What is the status of the problem---are the geodesics on a torus known?

I am interested---what are the parameters of the torus you used, and how dependant are your results on this? That is, two numbers specify a torus---the radius of the two circles. (Or, one can use a complex plane and specify a torus with a modulous and a phase.) Either way, is there a choice of r1 and r2 such that, for two points on the equator, the equator is always the shortest path? (This might be a trivial question---I could imagine that it is the case when one of the radii shrinks to zero.)

This reminds me of a problem I did as an undergraduate---taking a limit of some function for a Math contest. I graphed it and said--"Look, it goes to one". And the math professors said "What if it goes to 0.999."

BenTheMan: Ignoring varations with elliptical cross sections, a torus is completely defined by two Radii: R and rho. rho is the radius of what might be called the edible part of the doughnut. R is the distance from the center of the hole to the center of the edible part. I suppose you could characterize a torus by a single parameter (the ratio of the radii), but it seems easier to think in terms of the two radii. I tend to assign rho a value of one, but my program will accommodate other values.

It is interesting that (for fixed rho) torus geometry approaches cylindrical geometry as R grows without bound, and approaches spherical geometry as R approaches zero.

Longitude (0 to 360) and Latitude (-180 to +180) are convenient for use as coordinates. I tend to use terms related to a sphere when describing a torus. There is an Outer Equator (Latitude = 0) and an Inner Equator (latitude = 180). There is a North & a South Polar Circle (Latitude = +90 & Latitude = -90).

There are two second order differential equations which specify torus geodesics. There is a constant of integration whose value controls which of various types of geodesics are generated by numeric solutions to the differential equations. There is no general analytic solution to these equations.

As for any geometric object, neither the describing parameters nor co-ordinate system used affect the basic geometry, although the choices made can make the programming and/or the computations more formidable.

Obviously rigorous proof is more reliable than computational results, but analytical solutions and proofs are often not available. I have tried searching the web, but have found no analytical proofs of various torus properties strongly supported by numerical results.

For a torus with R = 3 and rho = 1, I obtained the following computational results for two points on the Outer Equator (rounded a bit).Attack angle = 41, Maximum latitude = 90, Equator crossing longitude = 104

Attack angle = 15, Maximum latitude = 30, Equator crossing longitude = 91

Attack angle = 5, Maximum latitude = 10, Equator crossing longitude = 90.128

Attack angle = 1.5, Maximum latitude = 3.0, Equator crossing longitude = 90.012

Attack angle = 0.410, Maximum latitude = 0.810, Equator crossing longitude = 90.001The above data is for geodesics which start on the Outer Equator (Longitude = 0 & Latitude = 0). The latitude for these geodesics reaches a maximum value, after which they return to the Equator. The Attack angle is the starting angle between the geodesic and the Outer Equator at longitude = 0 & latitude = 0. The Equator crossing longitude defines the crossing point, at which Latitude returns to zero.

Note that the crossing longitude seems to approach some value near 90 degrees as the attack angle and maximum latitude decrease. Note that the computations clearly indicate that there is some limiting value near 90. Surely that value is greater than 85 degrees. It is tempting to believe that 90 is the actual value.

For points on the Equator closer than that limiting value, the Equator is the shortest distance and there is no geodesic of the type described above. For points farther apart than that limiting value, there is a shorter geodesic than the Equator between the two points. It seems obvious (but perhaps not provable) that for two points separated by that limiting value, the Outer Equator and another geodesic have he same path length between the points.