Union of Spherical Caps

Date: 09/10/2001 at 09:07:06
From: Phil Koken
Subject: Partial volume of overlapping spheres

Dr. Math, 

Suppose I have two spheres of radius r1 and r2 respectively. 
Suppose that they partly overlap. What's the formula and its 
derivation for the overlapping volume ? 

I need this for some research I am doing in which a sub-problem is 
the overlapping volume of spheres drawn by different observers. 

Thanks a lot!
Phil

Date: 09/10/2001 at 11:17:06
From: Doctor Rob
Subject: Re: Partial volume of overlapping spheres

Thanks for writing to Ask Dr. Math, Phil.

What you have is the union of two spherical caps. For the formula
for the volume of a spherical cap, see the following web page from
our Frequently Asked Questions (FAQ) area:

 http://mathforum.org/dr.math/faq/formulas/faq.sphere.html#spherecap   

To use the formula, you need to know the radius of the circle of
intersection, and the heights of the two caps. These will depend on 
the distance d between the centers of the two spheres. If the cap of 
the sphere with radius r1 has height h1, and the cap of the sphere 
with radius r2 has height h2, then one can compute that

   h1 = (r2^2-[r1-d]^2)/(2*d),
   h2 = (r1^2-[r2-d]^2)/(2*d),

and the radius r3 of the common circle is

   r3 = sqrt([(r1+r2)^2-d^2]*[d^2-(r1-r2)^2])/(2*d).

Now you can apply the formula.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/