Localized Geometric Query Problems

TL;DR

This paper introduces a new class of geometric query problems focused on preprocessing sets of objects in the plane to efficiently find the largest circle containing a query point without intersecting the set.

Contribution

It defines and studies the problem of preprocessing geometric sets for efficient largest circle queries that contain a point but avoid the set, covering point sets and polygon boundaries.

Findings

Developed algorithms for point set queries

Extended methods to polygon boundary sets

Achieved efficient query times for the problem

Abstract

A new class of geometric query problems are studied in this paper. We are required to preprocess a set of geometric objects $P$ in the plane, so that for any arbitrary query point $q$, the largest circle that contains $q$ but does not contain any member of $P$, can be reported efficiently. The geometric sets that we consider are point sets and boundaries of simple polygons.