Circle around three given circles

Question

Given three arbitrary, non-overlapping circles, I need to find the smallest circle that encloses all three circles. I need a formula for the non-trivial case, where the enclosing circle touches all three circles.

The radii and the distances and angles between the circles are known. I used the fact that the distances from the center of each circle to the center of the enclosing circle plus the corresponding radius must be equal to the radius of the enclosing circle.

I found a very complex polynomial of degree 4 that assumingly should give me solutions for the radius of the enclosing circle, but I expected a solvable equation that directly gives me the radius.

Any ideas or solutions are appreciated.

Answer 1

I needed an answer to the same problem to assemble bounding circles in collision detection.

so here is how I do it

1. get the centroid G of the triangle (O1, O2, O3) made of the 3 circle centres
2. extend O1 by r1 in direction GO1> call it E1
3. do the same for O2 -> E2 (direction GO2>)
4. and O3 -> E3
5. get the circumcenter of triangle (E1, E2, E3) 5.