On existence of a compatible triangulation with the double circle order type

TL;DR

This paper proves that the double circle order type and some of its generalizations can be compatibly triangulated with any other order types sharing the same number of points and convex hull edges, advancing a conjecture in computational geometry.

Contribution

It establishes the existence of compatible triangulations for the double circle order type and certain generalizations, supporting a conjecture in the field.

Findings

Double circle order type has a compatible triangulation with any other order type with same point count.

Supports a specific case of a broader conjecture in computational geometry.

Generalizations of the double circle order type also admit compatible triangulations.

Abstract

We show that the "double circle" order type and some of its generalizations have a compatible triangulation with any other order types with the same number of points and number of edges on convex hull, thus proving another special case of the conjecture in Aichholzer (2003).