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PreTriangulations and Liftable Complexes # Oswin Aichholzer 1 , Franz Aurenhammer 2 , Thomas Hackl 1

Summary: Pre­Triangulations and Liftable Complexes #
Oswin Aichholzer 1 , Franz Aurenhammer 2 , Thomas Hackl 1
1 Institute for Software Technology
2 Institute for Theoretical Computer Science
Graz University of Technology, Graz, Austria
We introduce the concept of pre­triangulations, a relaxation of triangulations that
goes beyond the frequently used concept of pseudo­triangulations. Pre­triangulations
turn out to be more natural than pseudo­triangulations in certain cases. We show that
pre­triangulations arise in three di#erent contexts: In the characterization of polygonal
complexes that are liftable to three­space in a strong sense, in flip sequences for general
polygonal complexes, and as graphs of maximal locally convex functions.
1 Introduction
Polygonal complexes in the plane have been objects of interest in combinatorial geometry
from various points of view. With the advent of computational geometry, it soon became
apparent that combinatorial and geometric properties of certain polygonal complexes prove
useful for structuring geometric data and designing e#cient algorithms. Classical examples
are line arrangements that arise as duals of finite point sets [9], Voronoi diagrams that capture


Source: Aurenhammer, Franz - Institute for Theoretical Computer Science, Technische Universität Graz


Collections: Computer Technologies and Information Sciences