Union of Spherical CapsDate: 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/ |
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