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Let two spheres of radii and be located along the x-axis centered at and , respectively. Not surprisingly, the analysis is very similar to the case of the circle-circle intersection. The equations of the two spheres are
(1)
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(2)
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(3)
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Multiplying through and rearranging give
(4)
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Solving for gives
(5)
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The intersection of the spheres is therefore a curve lying in a plane parallel to the -plane at a single -coordinate. Plugging this back into (◇) gives
(6)
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(7)
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(8)
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(9)
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The volume of the three-dimensional lens common to the two spheres can be found by adding the two spherical caps. The distances from the spheres' centers to the bases of the caps are
(10)
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(11)
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so the heights of the caps are
(12)
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(13)
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The volume of a spherical cap of height for a sphere of radius is
(14)
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Letting and and summing the two caps gives
(15)
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(16)
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This expression gives for as it must. In the special case , the volume simplifies to
(17)
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In order for the overlap of two equal spheres to equal half the volume of each individual sphere, the spheres must be separated by a distance
(18)
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(19)
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(20)
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(OEIS A133749) times their radius, where is a polynomial root.
The surface area of the sphere that lies inside the sphere is equal to the great circle of the sphere , provided that (Kern and Blank 1948, p. 97).