math - How do I detect intersections between a circle and any other circle in the same plane?

Question

I'm looking for an algorithm to detect if a circle intersects with any other circle in the same plane (given that there can be more than one circle in a plane).

One method I have found is to do the separating axis test. It says:

Two objects don't intersect if you can find a line that separates the two objects, i.e. a line such that all objects or points of an object are on different sides of the line.

However, I don't know how to apply this method to my case.

Can anybody help me?

Answer 1

Two circles intersect if, and only if, the distance between their centers is between the sum and the difference of their radii. Given two circles (x0, y0, R0) and (x1, y1, R1), the formula is as follows:

ABS(R0 - R1) <= SQRT((x0 - x1)^2 + (y0 - y1)^2) <= (R0 + R1)

Squaring both sides lets you avoid the slow SQRT, and stay with ints if your inputs are integers:

(R0 - R1)^2 <= (x0 - x1)^2 + (y0 - y1)^2 <= (R0 + R1)^2

Since you need only a yes/no test, this check is faster than calculating the exact intersection points.

The above solution should work even for the "one circle inside the other" case.