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Re: Calculation of the surface after intersection of two
*To*: mathgroup at smc.vnet.net
*Subject*: [mg100884] Re: [mg100863] Calculation of the surface after intersection of two
*From*: DrMajorBob <btreat1 at austin.rr.com>
*Date*: Wed, 17 Jun 2009 04:37:32 -0400 (EDT)
*References*: <200906170153.VAA25938@smc.vnet.net>
*Reply-to*: drmajorbob at bigfoot.com
There's a mathematical formula for this:
http://mathworld.wolfram.com/Sphere-SphereIntersection.html
which gives, I think,
Clear[volume]
volume[R_, r_, d_] =
Pi/(12 d) (R + r - d)^2 (d^2 + 2 d r - 3 r^2 + 2 d R + 6 r R -
3 R^2);
volume[6/10, 6/10, 1]
N@%
(17 \[Pi])/1500
0.0356047
Here are some of the results from the article, derived in Mathematica:
volume[R, R, d] // Simplify
1/12 \[Pi] (d - 2 R)^2 (d + 4 R)
volume[R, r, r + R]
0
eqn = volume[R, R, k R] == (4/3 Pi R^3)/2 // Simplify
(8 - 12 k + k^3) R == 0
Array[N@Root[eqn[[1, 1]], #] &, 3]
{-3.75877, 0.694593, 3.06418}
d can't be negative, and if d >= 2 the spheres don't intersect, so only
the second root is valid.
Bobby
On Tue, 16 Jun 2009 20:53:32 -0500, Horacius ReX <horacius.rex at gmail.com>
wrote:
> Hi,
>
> I have two spheres of radius 0.6 whose origins are separated a
> distance of 1. So the spheres overlap and I want to calculate the
> total surface now.
>
> For that purpose I started to calculate the surface of the separated
> spheres, which is trivial and I can do by hand, but after they
> intersect or overlap, how can I tell the program to calculate the
> surface ?
>
> Thanks in advance
>
--
DrMajorBob at bigfoot.com
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