Circle-Circle Intersection

Two circles may intersect in two imaginary points, a single degenerate point, or two distinct points.

The intersections of two circles determine a line known as the radical line. If three circles mutually intersect in a single point, their point of intersection is the intersection of their pairwise radical lines, known as the radical center.

Let two circles of radii  and  and centered at  and  intersect in a region shaped like an asymmetric lens. The equations of the two circles are

			(1)
			(2)

Combining (1) and (2) gives

(3)

Multiplying through and rearranging gives

(4)

Solving for  results in

(5)

The chord connecting the cusps of the lens therefore has half-length  given by plugging  back in to obtain

			(6)
			(7)

Solving for  and plugging back in to give the entire chord length  then gives

			(8)
			(9)

This same formulation applies directly to the sphere-sphere intersection problem.

To find the area of the asymmetric "lens" in which the circles intersect, simply use the formula for the circular segment of radius  and triangular height 

(10)

twice, one for each half of the "lens." Noting that the heights of the two segment triangles are

			(11)
			(12)

The result is

			(13)
			(14)

The limiting cases of this expression can be checked to give 0 when  and

			(15)
			(16)

when , as expected.

In order for half the area of two unit disks () to overlap, set  in the above equation

(17)

and solve numerically, yielding  (OEIS A133741).

If three symmetrically placed equal circles intersect in a single point, as illustrated above, the total area of the three lens-shaped regions formed by the pairwise intersection of circles is given by

(18)

Similarly, the total area of the four lens-shaped regions formed by the pairwise intersection of circles is given by

(19)

REFERENCES:

Sloane, N. J. A. Sequence A133741 in "The On-Line Encyclopedia of Integer Sequences."