Sphere-Sphere Intersection

A sphere-sphere intersection is the set of points common to two spheres. Let their positive radii be and and their centers lie along the x-axis at and , respectively, with . The circle and spherical cap formulas below assume . For , tangency occurs when or , and the spheres meet at one point. If and , they coincide. In all other cases outside the proper-overlap range, their surfaces do not intersect. Not surprisingly, the analysis is very similar to the case of the circle-circle intersection. The equations of the two spheres are

Combining (1) and (2) gives

Multiplying through and rearranging give

Solving for gives

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 (1) gives

which is a circle with radius

The volume common to the two enclosed closed balls can be found by adding the two spherical caps. The signed offsets from the spheres' centers to the bases of the caps are

so the heights of the caps are

The volume of a spherical cap of height for a sphere of radius is

Letting and and summing the two caps gives

This expression gives for as it must. For , the common volume is 0. For , including concentric or coincident spheres, it is the volume of the smaller closed ball, . In the special case and , the volume simplifies to

In order for the overlap of two equal closed balls to equal half the volume of each individual closed ball, their centers must be separated by a distance

(OEIS A133749) times their radius, where is a polynomial root.

The surface area of the portion of the sphere of radius lying inside the closed ball bounded by the sphere of radius is , the area of a disk bounded by a great circle of the latter sphere, provided that and (Kern and Bland 1948, p. 97).

A related construction is the intersection of eight closed balls of common radius whose centers are the vertices of a cube of edge length . For centers , with all eight choices of signs, the common region is given by

Its volume is

(Månsson 1981, pp. 160-161).