Intersection of two circles

Written by Paul Bourke
April 1997

The following note describes how to find the intersection point(s) between two circles on a plane, the following notation is used. The aim is to find the two points P3 = (x3, y3) if they exist.

First calculate the distance d between the center of the circles. d = || P1 - P0 ||. If d > r0 + r1 then there are no solutions, the circles are separate. Also, if d < || r0 - r1 || then there are no solutions because one circle is contained within the other.

Considering the two triangles P0P2P3 and P1P2P3 we can write

a2 + h2 = r02 and b2 + h2 = r12

Using d = a + b we can solve for a,

a = (r02 - r12 + d2 ) / (2 d)

It can be readily shown that this reduces to r0 when the two circles touch at one point, ie: d = r0 + r1

Solve for h by substituting a into the first equation, h2 = r02 - a2

So

P2 = P0 + a ( P1 - P0 ) / d

And finally, P3 = (x3,y3) in terms of P0 = (x0,y0), P1 = (x1,y1) and P2 = (x2,y2), is

x3 = x2 +- h ( y1 - y0 ) / d

y3 = y2 -+ h ( x1 - x0 ) / d

Contribution: C source code example by Tim Voght.