Warp wrote:
>   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.

A few questions: ARe you talking about gravity in the surronding space 
(i.e., flying a space ship around this planet) or are you talking about 
gravity on the surface?

Just off the top of my head, it doesn't seem like it would be terribly 
difficult to solve the gravity equation analytically (integrating either 
over cylinders or disks-with-holes) at a bunch of points, and then 
interpolate between those points for other points, assuming you're talking 
about the outer-space gravity.  Even on the surface, it would seem gravity 
has to change smoothly and it should be pretty obvious where the inflection 
points would be.

The way you're talking about slicing things up, you *are* talking about 
integrating the volume to find the gravity in a particular direction. You're 
just trying to figure out how to do an integration with easier shapes.

A quick google turns up http://www.mathpages.com/home/kmath402/kmath402.htm
But you probably already did that.

It also seems if you made *enough* test point masses, worrying about how to 
subdivide it wouldn't be worthwhile. If your torus has radius 1, and you 
slice it into 10,000 test masses, isn't that going to give you enough 
accuracy? Use the digitalness of your compuations to advantage. Put mass in 
a sphere if the center of the sphere is inside the torus, and calculate from 
there.

I guess it depends on if you want to pre-calculate the field or whether 
you're trying to do this without actually storing too much data.  I usually 
work on the "store lots of data" sorts of problems, so my spidey-sense is 
off here.

Just don't try to work out the super-massive rotating torus gravity. :-)

-- 
Darren New, San Diego CA, USA (PST)
  "How did he die?"   "He got shot in the hand."
     "That was fatal?"
          "He was holding a live grenade at the time."

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