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Re: Smallest enclosing circle

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  • Subject: [mg50546] Re: Smallest enclosing circle
  • From: Matt Pharr <matt at>
  • Date: Thu, 9 Sep 2004 05:19:22 -0400 (EDT)
  • References: <cfq404$r9m$>
  • Sender: owner-wri-mathgroup at

[No mathematica code below, but some useful background.]

This problem actually comes up frequently on,
frequently enough that there's a FAQ entry there about it.  A lot of one's
intuition about this problem is often wrong.

To wit:


1.5 How can the smallest circle enclosing a set of points be found?

This circle is often called the minimum spanning circle. It can be computed
in O(n log n) time for n points.  The center lies on the furthest point
Voronoi diagram. Computing the diagram constrains the search for the
center. Constructing the diagram can be accomplished by a 3D convex hull
algorithm; [...]

In fact, the smallest circle can be computed in expected time O(n) by first
applying a random permutation of the points. The general concept applies to
spheres as well (and to balls and ellipsoids in any dimension).  The
algorithm uses a linear programming approach, proved by Emo Welzl in
[Wel91]. Code developed by Bernd Gaertner is available (GNU General Public
License) at

Matt Pharr    matt at    <URL:>
In a cruel and evil world, being cynical can allow you to get some
entertainment out of it. --Daniel Waters

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