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Some Counterexamples for Compatible Triangulations

Cody Barnson, Dawn Chandler, Qiao Chen, Christina Chung, Andrew Coccimiglio, Sean La, Lily Li, Aïna Linn, Anna Lubiw, Clare Lyle, Shikha Mahajan, Gregory Mierzwinski, Simon Pratt, Yoon Su Matthias Yoo, Hongbo Zhang, Kevin Zhang

TL;DR

The paper investigates whether any two planar point sets with equal size $n$ and hull size $h$ admit compatible triangulations. It constructs counterexamples to several strengthened forms, including cases with collinear points, and analyzes how fixing partial mappings or forcing edges can destroy compatibility. It shows that for point sets with at most two interior points compatibility always exists (even with collinearities), and examines the role of Steiner points when a mapping is specified, demonstrating that two Steiner points can suffice in some cases but one is not always enough. The results highlight that the existence of compatible triangulations depends sensitively on mappings and edge constraints and raise algorithmic questions about detecting compatibility with Steiner points.

Abstract

We consider the conjecture by Aichholzer, Aurenhammer, Hurtado, and Krasser that any two points sets with the same cardinality and the same size convex hull can be triangulated in the "same" way, more precisely via \emph{compatible triangulations}. We show counterexamples to various strengthened versions of this conjecture.

Some Counterexamples for Compatible Triangulations

TL;DR

The paper investigates whether any two planar point sets with equal size and hull size admit compatible triangulations. It constructs counterexamples to several strengthened forms, including cases with collinear points, and analyzes how fixing partial mappings or forcing edges can destroy compatibility. It shows that for point sets with at most two interior points compatibility always exists (even with collinearities), and examines the role of Steiner points when a mapping is specified, demonstrating that two Steiner points can suffice in some cases but one is not always enough. The results highlight that the existence of compatible triangulations depends sensitively on mappings and edge constraints and raise algorithmic questions about detecting compatibility with Steiner points.

Abstract

We consider the conjecture by Aichholzer, Aurenhammer, Hurtado, and Krasser that any two points sets with the same cardinality and the same size convex hull can be triangulated in the "same" way, more precisely via \emph{compatible triangulations}. We show counterexamples to various strengthened versions of this conjecture.

Paper Structure

This paper contains 7 sections, 1 theorem, 13 figures.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Collinear Points
  4. Compatible triangulations of small point sets with collinearities
  5. Specifying part of the mapping
  6. Specifying some of the edges
  7. Steiner points when the mapping is specified

Key Result

Theorem 1

If two points sets of size $n$ (possibly with collinear points) have 1 or 2 internal points, then they have a compatible triangulation.

Figures (13)

  • Figure 1: The counterexample to Conjecture \ref{['conj:compatible-exists']} in case of collinear points from Aichholzer et al. AICHHOLZER20033.
  • Figure 2: Point sets $P$ and $Q$.
  • Figure 3: $T_{P}$, $T_{Q_1}$, and $T_{Q_2}$.
  • Figure 4: Another counterexample to Conjecture \ref{['conj:compatible-exists']} for the case of collinear points. Observe that the left hand triangulation is unique and point $x$ has degree 6, but no point on the right can have degree 6 in any triangulation.
  • Figure 5: With the indicated mapping of convex hull points, these point sets have no compatible triangulation. Note that the triangulations are unique.
  • ...and 8 more figures

Theorems & Definitions (8)

  • Conjecture 1
  • Claim 1
  • Claim 2
  • proof
  • Theorem 1
  • proof : Proof of Theorem \ref{['thm:2-internal']}
  • Claim 3
  • proof