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Given a set $S$ of $n$ static points and a free point $p$ in the Euclideanplane, we study a new variation of the minimum enclosing circle problem, inwhich a dynamic weight that equals to the reciprocal of the distance from thefree point $p$ to the undetermined circle center is included. In this work, weprove the optimal solution of the new problem is unique and lies on theboundary of the farthest-point Voronoi diagram of $S$, once $p$ does notcoincide with any vertex of the convex hull of $S$. We propose a tree structureconstructed from the boundary of the farthest-point Voronoi diagram and use thehierarchical relationship between edges to locate the optimal solution. Theplane could be divide into at most $3n-4$ non-overlapping regions. When $p$lies in one of the regions, the optimal solution locates at one node or lies onthe interior of one edge in the boundary of the farthest-point Voronoi diagram.Moreover, we apply the new variation to calculate the maximum displacement ofone point $p$ under the condition that the displacements of points in $S$ arerestricted in 2D rigid motion.

Minimum Enclosing Circle of a Set of Static Points with Dynamic Weight from One Free Point