algorithm - Circle-circle intersection points

Question

How do I calculate the intersection points of two circles. I would expect there to be either two, one or no intersection points in all cases.

I have the x and y coordinates of the centre-point, and the radius for each circle.

An answer in python would be preferred, but any working algorithm would be acceptable.

Answer 1

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 r

₀ 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/