Point enclosure problem for homothetic polygons - Akram, Waseem; Saxena, Sanjeev (Theoretical computer science, 2025)
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(nlog⁡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.