Le 04/03/2011 03:17, Warp a écrit :
>   I explained it in my first post: I want to make a numerical approximation
> of the gravity of a torus, and a way of doing that is to create point masses
> inside it.
> 
>   Perhaps you should read my first post again. I *want* to get an even
> distribution of mass. That's my question: What formula should I use to
> get it? I explained the approach in detail, it's just the specifics that
> I'm after.
> 

Simplest distribution: along the major radius circle. Kind of 1D
distribution, evenly spaced with same masse along a circle. Easy.

Now, if you intend to refine that point inside the minor disc on the
perpendicular plane, you are in trouble for an "all the same mass"
points scheme. But that should not stop you from substituing for
instance the central point with seven points (hexagon pattern with
center), as long as you adjust the mass of each point according to its
distance from the main axis of the torus.

What can be done for 1--> 7 can be pushed further with any triangular
tiling of the circle. The points would not be even spaced, nor have all
the same mass, but it would work.

If you really want even spaced points of same mass... put a high
resolution grid (of regular tetrahedron instead of cube if you care) and
generate two set of points:
 * first set is made of the center of tetrahedrons completely in the torus
 * second set is made the center of tetrahedrons with at least one part
inside the torus.

Sets 1 & 2 would converge to the same when the resolution is increased,
so you can probably choose a basic test: if the center of the
tetrahedron is inside the torus, it has full mass.

About the mass of each points: the density * volume of torus divided by
the number of points!


For symmetry, it would be nice to have the hexagonal paving of the grid
to align with the plane of the torus.

Look at http://en.wikipedia.org/wiki/Sphere_packing
in particular, the regular packing. each sphere's center is a point of
your grid (cubic close packing/face centered cubic). Sample the torus
and you are done.



-- 
Software is like dirt - it costs time and money to change it and move it
around.

Just because you can't see it, it doesn't weigh anything,
and you can't drill a hole in it and stick a rivet into it doesn't mean
it's free.

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