On joint triangulations of two sets of points in the plane

TL;DR

This paper investigates conditions for joint triangulations of two point sets in the plane, proposing necessary conditions, a conjecture on their sufficiency, and an efficient algorithm for polygons.

Contribution

It introduces two necessary conditions for joint triangulations, conjectures their sufficiency, and provides an $O(n^3)$ dynamic programming algorithm for simple polygons.

Findings

Necessary conditions for joint triangulation established

Algorithm for joint triangulation of polygons developed

Testing conditions can be done in $O(n^3)$ time

Abstract

In this paper, we establish two necessary conditions for a joint triangulation of two sets of $n$ points in the plane and conjecture that they are sufficient. We show that these necessary conditions can be tested in $O(n^3)$ time. For the problem of a joint triangulation of two simple polygons of $n$ vertices, we propose an $O(n^3)$ time algorithm for constructing a joint triangulation using dynamic programming.