Sphere-Sphere Intersection

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SphereSphereIntersection

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

[画像: 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 = [画像:R^2-x^2=R^2-((d^2-r^2+R^2)/(2d))^2]

(6)

= [画像:(4d^2R^2-(d^2-r^2+R^2)^2)/(4d^2),]

(7)

which is a circle with radius

a = [画像:1/(2d)sqrt(4d^2R^2-(d^2-r^2+R^2)^2)]

(8)

= [画像: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 = [画像:R-d_1=((r-R+d)(r+R-d))/(2d)]

(12)

h_2 = [画像: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)

= [画像:(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).

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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

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

Sphere-Sphere Intersection

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