Some Counterexamples for Compatible Triangulations
Cody Barnson, Dawn Chandler, Qiao Chen, Christina Chung, Andrew, Coccimiglio, Sean La, Lily Li, A\"ina Linn, Anna Lubiw, Clare Lyle, Shikha, Mahajan, Gregory Mierzwinski, Simon Pratt, Yoon Su Matthias Yoo, Hongbo, Zhang, Kevin Zhang
TL;DR
This paper presents counterexamples to a conjecture that any two point sets with identical size and convex hull can be triangulated compatibly, challenging assumptions in geometric triangulation theory.
Contribution
It provides the first known counterexamples to several strengthened versions of the compatible triangulation conjecture.
Findings
Counterexamples disprove certain strengthened conjectures
Challenges assumptions about triangulation compatibility
Advances understanding of geometric triangulation limitations
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.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Point processes and geometric inequalities · Commutative Algebra and Its Applications