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

## TL;DR

This paper introduces a novel variation of the minimum enclosing circle problem involving a dynamic weight based on a free point's position, providing a unique optimal solution framework using farthest-point Voronoi diagrams.

## Contribution

It proves the uniqueness of the optimal solution and develops a tree-based method to locate it within the farthest-point Voronoi diagram structure.

## Key findings

- Optimal solution is unique and on the boundary of the farthest-point Voronoi diagram.
- The plane is divided into at most 3n-4 regions for solution localization.
- Application to maximum displacement calculation under rigid motion constraints.

## Abstract

Given a set $S$ of $n$ static points and a free point $p$ in the Euclidean plane, we study a new variation of the minimum enclosing circle problem, in which a dynamic weight that equals to the reciprocal of the distance from the free point $p$ to the undetermined circle center is included. In this work, we prove the optimal solution of the new problem is unique and lies on the boundary of the farthest-point Voronoi diagram of $S$, once $p$ does not coincide with any vertex of the convex hull of $S$. We propose a tree structure constructed from the boundary of the farthest-point Voronoi diagram and use the hierarchical relationship between edges to locate the optimal solution. The plane 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 on the interior of one edge in the boundary of the farthest-point Voronoi diagram. Moreover, we apply the new variation to calculate the maximum displacement of one point $p$ under the condition that the displacements of points in $S$ are restricted in 2D rigid motion.