Journal of the Franklin Institute 344 (2007) 929–940 On the geometry of the smallest circle enclosing a finite set of points Lance D. Drager à , Jeffrey M. Lee, Clyde F. Martin Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409 1042, USA Received 8 January 2005; received in revised form 31 December 2006; accepted 16 January 2007 Abstract A number of numerical codes have been written for the problem of finding the circle of smallest radius in the Euclidean plane that encloses a finite set P of points, but these do not give much insight into the geometry of this circle. We investigate geometric properties of the minimal circle that may be useful in the theoretical analysis of applications. We show that a circle C enclosing P is minimal if and only if it is rigid in the sense that it cannot be translated while still enclosing P. We show that the center of the minimal circle is in the convex hull of P. We use this rigidity result and an analysis of the case of three points to find sharp estimates on the diameter of the minimal circle in terms of the diameter of P. r 2007 Published by Elsevier Ltd. on behalf of The Franklin Institute. Keywords: Smallest enclosing circle 1. Introduction In this paper we study some geometric properties of the circle of smallest radius enclosing a finite set of points in the Euclidean plane. The results of this paper arose when considering a problem that has roots in a problem of communication between mobile robots. The particular problem we were studying was the following. Suppose we have p points in the plane that are pairwise at least r units apart. What is the radius of the minimal circle that encloses them? The dual problem is also of ARTICLE IN PRESS www.elsevier.com/locate/jfranklin 0016-0032/$32.00 r 2007 Published by Elsevier Ltd. on behalf of The Franklin Institute. doi:10.1016/j.jfranklin.2007.01.003 à Corresponding author. Tel.: +1 806 742 2580; fax: +1 806 742 1112. E-mail addresses: lance.drager@ttu.edu (L.D. Drager), jeffrey.lee@ttu.edu (J.M. Lee), clyde.f.martin@ttu.edu (C.F. Martin).