Minimum enclosing circle of a set of fixed points and a mobile point

Abstract

In this paper we study the variations in the center and the radius of the minimum enclosing circle (MEC) of a set of fixed points and one mobile point, moving along a straight line ℓ. Given a set S of n points and a line ℓ in ℝ2, we completely characterize the locus of the center of MEC of the set S ∪ {p}, for all p ∈ ℓ. We show that the locus is a continuous and piecewise differentiable linear function, and each of its differentiable piece lies either on the edges of the farthest-point Voronoi diagram of S, or on a line segment parallel to the line ℓ. Moreover, we prove that the locus can have at most O(n) differentiable pieces and can be computed in linear time, given the farthest-point Voronoi diagram of S. © 2011 Springer-Verlag.