On 3/3/2011 6:25 PM, Warp wrote:
>
>> If the torus isn't too `skinny' you could just uniformly sample a bunch
>> of points from its bounding box and retain only those which lie inside
>> the torus.
>
>    There's a minor problem with that approach which might skew the result
> in some cases. Basically, you are dividing the torus into small cubes and
> putting the points at the center of each cube. However, some of the cubes
> are not cubes because they are cut by the surface of the torus. In these
> cases the point masses do not correspond to the volumes of the clipped
> cubes.

Of course, but Darren's point holds in this case.  The marching cubes 
approach fixes this if you're really concerned.


>    (There's also the problem that some of the points will get too close
> to the surface that way. If the location we are measuring the gravity
> from happens to be too close to such a point, the result will be way too
> high. In practice you have a small black hole near the surface of the
> torus, and the test location is too close to it, skewing the result.)

I would think that any approach based on approximating the torus as a 
bunch of point masses would have this problem.


>    I think it's just easier to scale the point masses according to a simple
> quadratic function. It's just the exact function I'm looking for.

Does just scaling the mass proportionally to a point's distance from the 
torus' axis not work here?  If you're looking for an exact equation 
based on the actual mass of the torus, just run a post-process 
normalization pass where you scale the mass of each point so that they 
sum to the total mass of the torus.

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