       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>

```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 ?
>