Constrained paths in the flip-graph of regular triangulations
type: Item
instance of: scholarly article
title: Constrained paths in the flip-graph of regular triangulations
zbMATH Open document ID: 1123.68132
author: Lionel Pournin
author: Thomas M. Liebling
published in: Computational Geometry
publication date: 2007-04-18T00:00:00Z
full work available at URL: PouLie Constrained paths in the flip-graph.pdf
review text: This note regards some special geometric operation that arises in many applications from mesh generation to algebraic geometry. A triangulation of a point set \(A\) is a simplicial complex \(T\) with a geometric realization \(| T |\) such that the (1) vertices of the simplices are points of \(A\), (2) the usual abstract properties of simplicial complexes are respected in the embedding, and (3) \(| T |\), as a subset of Euclidean space, is precisely the convex hull of \(A\). Geometric bistellar flips (also called geometric Pachner moves) are simple geometric operations that replace a subcomplex inside the triangulation by a new simplicial complex that yields a new triangulation. Note that the set of vertices \(A\) is fixed and thus the number of distinct triangulations is finite. A triangulation is regular when it is the projection of a convex triangulated surface of one higher dimension. A classical example is a Delaunay triangulation. It has been known for some time now that the set of all regular triangulations is connected by flips (a result going back to 1990 due to Gel'fand Kapranov and Zelevinsky). The present note shows that in fact any pair of regular triangulations is connected by a sequence of finitely many flips where none of their common faces are destroyed. As a consequence it follows the connectivity of the subgraph induced by all regular triangulations that share the same vertex set.
Mathematics Subject Classification ID: 68U05
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 5144508
zbMATH Keywords: regular triangulations
zbMATH Keywords: secondary polytopes
zbMATH Keywords: flips
MaRDI profile type: Publication
OpenAlex ID: W2157015489
cites work: The number of geometric bistellar neighbors of a triangulation
cites work: Triangulations. Structures for algorithms and applications
cites work: Tetrahedrizing point sets in three dimensions
cites work: Incremental topological flipping works for regular triangulations
cites work: Q4711942
cites work: Discriminants, resultants, and multidimensional determinants
cites work: Construction of three-dimensional Delaunay triangulations using local transformations
cites work: Q4194442
cites work: A New Approach to the Analysis of Free Rotations of Rigid Bodies
cites work: Three-dimensional distinct element simulation of spherocylinder crystallization
cites work: A point set whose space of triangulations is disconnected
cites work: Triangulations with very few geometric bistellar neighbors
cites work: Non-connected toric Hilbert schemes
DOI: 10.1016/J.COMGEO.2006.07.001
description: scientific article; zbMATH DE number 5144508