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Adapting (Pseudo)-Triangulations with a Near-Linear Number of Edge Flips

  • Oswin Aichholzer
  • Franz Aurenhammer
  • Hannes Krasser
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2748)

Abstract

In geometric data processing, structures that partition the geometric input, as well as connectivity structures for geometric objects, play an important role. A versatile tool in this context are triangular meshes, often called triangulations; see e.g., the survey articles [6, 12, 5]. A triangulation of a finite set S of points in the plane is a maximal planar straight-line graph that uses all and only the points in S as its vertices. Each face in a triangulation is a triangle spanned by S. In the last few years, a relaxation of triangulations, called pseudo-triangulations (or geodesic triangulations), has received considerable attention. Here, faces bounded by three concave chains, rather than by three line segments, are allowed. The scope of applications of pseudo-triangulations as a geometric data stucture ranges from ray shooting [10, 14] and visibility [25, 26] to kinetic collision detection [1, 21, 22], rigidity [32, 29, 15], and guarding [31]. Still, only very recently, results on the combinatorial properties of pseudo-triangulations have been obtained. These include bounds on the minimal vertex and face degree [20] and on the number of possible pseudo-triangulations [27, 3]. The usefulness of (pseudo-)triangulations partially stems from the fact that these structures can be modified by constant-size combinatorial changes, commonly called flip operations. Flip operations allow for an adaption to local requirements, or even for generating globally optimal structures [6, 12]. A classical result states that any two triangulations of a given planar point set can be made to coincide by applying a quadratic number of edge flips; see e.g. [16, 19]. A similar result has been proved recently for the class of minimum pseudo-triangulations [8, 29].

Keywords

Computational Geometry Simple Polygon Internal Edge Optimal Triangulation Flip Operation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Franz Aurenhammer
    • 1
  • Hannes Krasser
    • 2
  1. 1.Institute for Software TechnologyGraz University of TechnologyGrazAustria
  2. 2.Institute for Theoretical Computer ScienceGraz University of TechnologyGrazAustria

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