Finding the smallest circle that encompasses other circles?

Question

If a circle is defined by the X, Y of it's center and a Radius, then how can I find a Circle that encompasses a given number of circles? A single circle that is the smallest possible circle to completely contain 2 or more circles of any size and location.

At first I tried just encompassing 2 circles by finding the midpoint of the centers and that being the midpoint of the new circle while the radius was equal to the half of the radius of the 2 initial circles and half the distance between their centers, but somehow it always turned out to be a little off. The problem always seemed to be a problem with finding the radius, but I have such a headache about this I can't make it work.

I don't necessarily need a method for finding a circle that encompasses 3 or more circles. I can find a circle that encompasses 2, take that circle and encompass it with another, and another, and the final circle should encompass all circles given throughout the steps.

Answer 1

There problem you have at hand is called Smallest enclosing sphere of spheres. I have written my thesis about it, see "Smallest enclosing ball of balls", ETH Zurich.

You can find a very efficient C++ implementation in the Computational Geometry Algorithms Library (CGAL) in package Bounding Volumes. (There is no need to use all of CGAL; just extract the required source and header files and you are up and running.)

Note: If you are looking for an algorithm to compute the smallest enclosing sphere of points only, there are other implementations out there, see this post.