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Re: Calculation of the surface after intersection of two
*To*: mathgroup at smc.vnet.net
*Subject*: [mg100904] Re: Calculation of the surface after intersection of two
*From*: "David W. Cantrell" <DWCantrell at sigmaxi.net>
*Date*: Thu, 18 Jun 2009 04:50:33 -0400 (EDT)
*References*: <200906170153.VAA25938@smc.vnet.net> <h1a9re$81h$1@smc.vnet.net>
drmajorbob at bigfoot.com wrote:
> 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
It seemed to me that Horacius was interested in surface area, rather than
volume. If that's correct, then he should be able to calculate what he
wants easily using the formula for the surface area of a spherical cap.
See, for example, <http://mathworld.wolfram.com/SphericalCap.html>, near
the end.
David
> 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
> >
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