Circle-circle intersection points

Intersection of two circles

Written by Paul Bourke

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.

• If d < |r₀ –
  r₁| then there are no solutions because one circle is
  contained within the other.

• If d = 0 and r₀ =
  r₁ then the circles are coincident and there are an
  infinite number of solutions.

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

Source: http://paulbourke.net/geometry/circlesphere/