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Circle-Circle Intersection


CircleIntersections

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.

CircleCircleIntersection

Let two circles of radii R and r and centered at (0,0) and (d,0) intersect in a region shaped like an asymmetric lens. The equations of the two circles are

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

Combining (1) and (2) gives

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

Multiplying through and rearranging gives

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

Solving for x results in

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

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

y^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)

Solving for y and plugging back in to give the entire chord length a=2y then gives

a=1/dsqrt(4d^2R^2-(d^2-r^2+R^2)^2)
(8)
=1/dsqrt((-d+r-R)(-d-r+R)(-d+r+R)(d+r+R)).
(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 R^' and triangular height d^'

 A(R^',d^')=R^('2)cos^(-1)((d^')/(R^'))-d^'sqrt(R^('2)-d^('2))
(10)

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

d_1=x=(d^2-r^2+R^2)/(2d)
(11)
d_2=d-x=(d^2+r^2-R^2)/(2d).
(12)

The result is

A=A(R,d_1)+A(r,d_2)
(13)
=r^2cos^(-1)((d^2+r^2-R^2)/(2dr))+R^2cos^(-1)((d^2+R^2-r^2)/(2dR))-1/2sqrt((-d+r+R)(d+r-R)(d-r+R)(d+r+R)).
(14)

The limiting cases of this expression can be checked to give 0 when d=R+r and

A=2R^2cos^(-1)(d/(2R))-1/2dsqrt(4R^2-d^2)
(15)
=2A(1/2d,R)
(16)

when r=R, as expected.

Circle-CircleIntersectionHalf

In order for half the area of two unit disks (R=1) to overlap, set A=piR^2/2=pi/2 in the above equation

 1/2pi=2cos^(-1)(1/2d)-1/2dsqrt(4-d^2)
(17)

and solve numerically, yielding d=0.8079455... (OEIS A133741).

Circle3Intersection

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

 A=pi-3/2sqrt(3).
(18)
Circle4Intersection

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

 A=2(pi-2).
(19)

See also

Borromean Rings, Brocard Triangles, Circle-Ellipse Intersection, Circle-Line Intersection, Circular Segment, Circular Triangle, Double Bubble, Goat Problem, Johnson's Theorem, Lens, Lune, Mohammed Sign, Moss's Egg, Radical Center, Radical Line, Reuleaux Triangle, Sphere-Sphere Intersection, Steiner Construction, Triangle Arcs, Triquetra, Venn Diagram, Vesica Piscis

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References

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

Cite this as:

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

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