# Is there a relation for when a circle intersects more than half the perimeter/circumference of another circle?

Is there some nice formula or algoritm for determining when a circle "hides"/intersects more than half of the perimeter of another circle, in a circle-circle interaction.

Example image: Two example circles on Wolframalpha

A way to either determine if more or less than half of the perimeter is "hidden" or a way to fully calculate the hidden perimeter length is desired.

radii, distance between circles, centers and intersection points (in a x-y coordinate system) are known.

• Let A be the point of circle $C_1$ closest to circle $C_2$, B and C the points of intersection of $C_1$ and $C_2$. Then the angle BAC should be smaller than $\pi/2$. – OR. Mar 20 '14 at 20:42
• In your algorithm you should contemplate degenerate cases. If B and C don't exist, then you must check if A is inside $C_2$. In this case the answer is that the whole circle $C_1$ is covered by $C_2$. Stuff like that. – OR. Mar 20 '14 at 20:44
• Yes, I meant to write that A is the point of $C_1$ closest to the center of $C_2$. – OR. Mar 20 '14 at 20:49
• Although it is not what I meant to write, the point of a circle closest to another circle can also be determined. The solution of that computation could be: One point, if the circles do not intersect and are not concentric, or if they are tangent to each other. Two points, if they intersect at two points. Or that all the point of $C_1$ are closest to $C_2$. This occurs when both circles are concentric. – OR. Mar 20 '14 at 20:53