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Sphere-Sphere Intersection


SphereSphereInterGraphicSphereSphereIntersection

Let two spheres of radii R and r be located along the x-axis centered at (0,0,0) and (d,0,0), respectively. Not surprisingly, the analysis is very similar to the case of the circle-circle intersection. The equations of the two spheres are

x^2+y^2+z^2=R^2
(1)
(x-d)^2+y^2+z^2=r^2.
(2)

Combining (1) and (2) gives

 (x-d)^2+(R^2-x^2)=r^2.
(3)

Multiplying through and rearranging give

 x^2-2dx+d^2-x^2=r^2-R^2.
(4)

Solving for x gives

 x=(d^2-r^2+R^2)/(2d).
(5)

The intersection of the spheres is therefore a curve lying in a plane parallel to the yz-plane at a single x-coordinate. Plugging this back into (◇) gives

y^2+z^2=R^2-x^2=R^2-((d^2-r^2+R^2)/(2d))^2
(6)
=(4d^2R^2-(d^2-r^2+R^2)^2)/(4d^2),
(7)

which is a circle with radius

a=1/(2d)sqrt(4d^2R^2-(d^2-r^2+R^2)^2)
(8)
=1/(2d)[(-d+r-R)(-d-r+R)(-d+r+R)(d+r+R)]^(1/2).
(9)

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

d_1=x
(10)
d_2=d-x,
(11)

so the heights of the caps are

h_1=R-d_1=((r-R+d)(r+R-d))/(2d)
(12)
h_2=r-d_2=((R-r+d)(R+r-d))/(2d).
(13)

The volume of a spherical cap of height h^' for a sphere of radius R^' is

 V(R^',h^')=1/3pih^('2)(3R^'-h^').
(14)

Letting R_1=R and R_2=r and summing the two caps gives

V=V(R_1,h_1)+V(R_2,h_2)
(15)
=(pi(R+r-d)^2(d^2+2dr-3r^2+2dR+6rR-3R^2))/(12d).
(16)

This expression gives V=0 for d=r+R as it must. In the special case r=R, the volume simplifies to

 V=1/(12)pi(4R+d)(2R-d)^2.
(17)

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

d=(x^3-12x+8)_2
(18)
=2sqrt(3)sin(2/9pi)-2cos(2/9pi)
(19)
=0.694592710...
(20)

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

The surface area of the sphere R that lies inside the sphere r is equal to the great circle of the sphere r, provided that r<=2R (Kern and Blank 1948, p. 97).


See also

Apple Surface, Circle-Circle Intersection, Cylinder-Sphere Intersection, Double Bubble, Lens, Reuleaux Tetrahedron, Space Division by Spheres, Sphere

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References

Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 97, 1948.Sloane, N. J. A. Sequence A133749 in "The On-Line Encyclopedia of Integer Sequences."

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Sphere-Sphere Intersection

Cite this as:

Weisstein, Eric W. "Sphere-Sphere Intersection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Sphere-SphereIntersection.html

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