Point enclosure problem for homothetic polygons

Abstract

In this paper, we investigate the following problem: “given a set $S$ of n homothetic polygons, preprocess $S$ to efficiently report all the polygons of $S$ containing a query point.” A set of polygons is said to be homothetic if each polygon can be obtained from any other polygon in the set using scaling and translation operations. The problem is a counterpart of the homothetic range search problem discussed by Chazelle and Edelsbrunner (1987) [9]. We show that after preprocessing a set of homothetic polygons with a constant number of vertices, queries can be answered in $O(\log n+k)$ optimal time, where k is the output size. The preprocessing takes $O(n\log n)$ space and time. We also study the problem in a dynamic setting where insertion and deletion operations are allowed. The results we obtain also hold for c-oriented triangles, where c is a fixed constant.