>    In another forum I was pondering about the gravity field of a torus
> (ie. imagine you had eg. a torus-shaped planet: What would be the direction
> and strength of gravity at different points on its surface?)
>
>    Two possibilities were suggested: The analytical (and thus exact) way,
> which would require solving a complicated volume (ie. triple) integral,
> and numerical approximation.
>
>    The numerical approximation is simpler in principle, but there are some
> difficulties.
>
>    One way to numerically approximate the result would be to fill the torus
> with point masses and then sum up their gravity on the tested location.
> If we assume that the torus has even density, filling it up with point
> masses is actually non-trivial.
>
>    There are two possibilities:
>
>    1) All the point masses have the same mass (ie. the total mass of the
> torus divided by the number of points) and are distributed evenly inside
> the torus. This is a bit problematic because coming up with an even
> distribution of points inside a torus is not easy. Basically it would
> mean that you would have to divide the torus into polyhedrons of the
> same volume, and put the point masses at their center. However, subdividing
> a torus into polyhedrons of the same volume is not trivial.
>
>    2) Instead, we subdivide the torus into polyhedrons of arbitrary size
> and scale the mass of the points in relation to the volume of the polyhedron.
> (In other words, the masses are scaled according to the local point
> density.)
>
>    The second option might be the easiest to do, as it allows for a simple
> distribution function for the points which does not need to be strictly
> even.
>
>    The easiest division of the torus would be to divide it into slices,
> each slice is divided into sections (each such section would thus
> effectively be a pyramid with the base on the surface of the torus and
> the apex on the central major-radius circle of the torus), and then each
> section into polyhedrons (from the base towards the apex of the pyramid).
>
>    One way to easily achieve this programmatically is to simply create
> points on the surface of toruses of increasing minor radius (the major
> radius being the same as the original torus we are dealing with).
>
>    The problem is deciding what the mass of these points should be. As said,
> they can't have equal mass because else you would end up with very uneven
> density (the torus would be significantly denser deep inside and less
> dense closer to the surface).
>
>    If we think about the points inside one of the slices, the masses should
> be, if I understand correctly, proportional to the square of the distance
> from the center. This gives you a disc of even density. (However, I'm not
> sure now if there should be a constant factor involved...)
>
>    However, that alone is not enough. There's another aspect that has to
> be considered: The points would be distributed less densely on the outer
> rim of the torus and more densely on the inner rim. Thus they have to be
> also scaled according to their distance from the center of the torus.
> I'm not exactly sure what the formula would be.
>
>    Since we want the entire torus to have a certain mass m, the sum of
> all the point masses have to be that. However, that's actually trivial:
> Simply use whatever values are most convenient to create the points, and
> then just divide by the sum of the original masses and multiply by m.
>
>    However, the exact formulas to determine the masses of the points is
> a bit fuzzy to me. Perhaps someone could help a bit?
>

Start with 2 points placed at the major radius.
Draw 2 circles of minor radius.

Calculate the 2D influence for those 2 circles.

Rotate around the major axis.

This assume that the density is constant.

It resolves to a static 2 body gravitic interaction.




Alain

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