Dominance for Containment Problems

TL;DR

This paper addresses geometric containment problems by reducing them to 3D dominance queries, providing solutions for various object types and extending to higher dimensions.

Contribution

It introduces a novel reduction of containment problems involving homothetic triangles to 3D dominance queries, enabling efficient solutions for multiple query object types.

Findings

Containment problems can be solved via 3D dominance queries.

Solutions extend to higher dimensions.

Applicable to line segments, circles, and trapezoids.

Abstract

In a containment problem, the goal is to preprocess a set of geometric objects so that, given a geometric query object, we can report all the objects containing the query object. We consider the containment problem where input objects are homothetic triangles and the query objects considered are line segments, circles, and trapezoids with bases parallel to either axis. We show that this problem can be solved using the 3-d query dominance problem. The solutions presented can also be extended for higher dimensions.