## Convexity Minimizes Pseudo-Triangulations (2002)

Venue: | Computational Geometry: Theory and Applications |

Citations: | 15 - 2 self |

### BibTeX

@ARTICLE{Aichholzer02convexityminimizes,

author = {Oswin Aichholzer and Franz Aurenhammer and Hannes Krasser and Bettina Speckmann},

title = {Convexity Minimizes Pseudo-Triangulations},

journal = {Computational Geometry: Theory and Applications},

year = {2002},

volume = {28},

pages = {158--161}

}

### Years of Citing Articles

### OpenURL

### Abstract

The number of minimum pseudo-triangulations is minimized for point sets in convex position.

### Citations

101 | A Combinatorial Approach to Planar Non-Colliding Robot Arm Motion Planning
- Streinu
- 2000
(Show Context)
Citation Context ...angulation is pointed, that is, its incident edges span a convex angle. In fact, minimum pseudo-triangulations can be characterized as maximal planar straight-line graphs where each vertex is pointed =-=[15]-=-. Pseudo-triangulations, also called geodesic triangulations, have received considerable attention in the last few years due to their applications to visibility [10, 11], ray shooting [6], kinetic col... |

96 |
Ray shooting in polygons using geodesic triangulations
- Chazelle, Edelsbrunner, et al.
(Show Context)
Citation Context ... is pointed [15]. Pseudo-triangulations, also called geodesic triangulations, have received considerable attention in the last few years due to their applications to visibility [10, 11], ray shooting =-=[6]-=-, kinetic collision detection [1, 8, 9], rigidity [15], and guarding [14]. While some of their interesting geometric and combinatorial properties have already been discovered [15, 13, 7, 5], many elem... |

86 | Topologically sweeping visibility complexes via pseudotriangulations
- Pocchiola, Vegter
- 1996
(Show Context)
Citation Context ...raphs where each vertex is pointed [15]. Pseudo-triangulations, also called geodesic triangulations, have received considerable attention in the last few years due to their applications to visibility =-=[10, 11]-=-, ray shooting [6], kinetic collision detection [1, 8, 9], rigidity [15], and guarding [14]. While some of their interesting geometric and combinatorial properties have already been discovered [15, 13... |

77 | Deformable free space tiling for kinetic collision detection
- Agarwal, Basch, et al.
- 2000
(Show Context)
Citation Context ...ations, also called geodesic triangulations, have received considerable attention in the last few years due to their applications to visibility [10, 11], ray shooting [6], kinetic collision detection =-=[1, 8, 9]-=-, rigidity [15], and guarding [14]. While some of their interesting geometric and combinatorial properties have already been discovered [15, 13, 7, 5], many elementary open questions still remain. For... |

70 |
Kinetic collision detection for simple polygons
- Kirkpatrick, Snoeyink, et al.
(Show Context)
Citation Context ...ations, also called geodesic triangulations, have received considerable attention in the last few years due to their applications to visibility [10, 11], ray shooting [6], kinetic collision detection =-=[1, 8, 9]-=-, rigidity [15], and guarding [14]. While some of their interesting geometric and combinatorial properties have already been discovered [15, 13, 7, 5], many elementary open questions still remain. For... |

45 | Expansive motions and the polytope of pointed pseudo-triangulations
- Rote, Santos, et al.
- 2003
(Show Context)
Citation Context ...10, 11], ray shooting [6], kinetic collision detection [1, 8, 9], rigidity [15], and guarding [14]. While some of their interesting geometric and combinatorial properties have already been discovered =-=[15, 13, 7, 5]-=-, many elementary open questions still remain. For example, little is known about the number of minimum pseudotriangulations a general point set S allows. In [12], the number of minimum pseudo-triangu... |

37 |
On the crossing number of complete graphs
- Aichholzer, Aurenhammer, et al.
- 2002
(Show Context)
Citation Context ... minimum pseudo-triangulations, which may be useful for checking the correctness of counting algorithms. A similar property is known for the number of crossings in complete geometric graphs, see e.g. =-=[3]-=-. No such observations exist for triangulations. Lemma 3 If the number h of extreme points of a set S is even then so is the number of minimum pseudotriangulations of S. Proof : Let V be the set of al... |

32 |
Enumerating Order Types for Small Point Sets with Applications. Order 19
- Aichholzer, Aurenhammer, et al.
- 2002
(Show Context)
Citation Context ... degree 2 or 3 has to exist. Treat p as in the proof of Lemma 2, which leads to a minimum pseudotriangulation of S \ {p}. The assertion now follows by induction. # Remark 3 The point set data base of =-=[2]-=- has been used by [5] to obtain the values of N n,h in Table 1. We exploited these values to slightly improve Theorem 1. For instance, for n # 7, we get the uniform bound N n,h # C h-2 4 n-h . Moreove... |

32 | Tight degree bounds for pseudo-triangulations of points. Computational Geometry: Theory and Applications
- Kettner, Kirkpatrick, et al.
(Show Context)
Citation Context ...10, 11], ray shooting [6], kinetic collision detection [1, 8, 9], rigidity [15], and guarding [14]. While some of their interesting geometric and combinatorial properties have already been discovered =-=[15, 13, 7, 5]-=-, many elementary open questions still remain. For example, little is known about the number of minimum pseudotriangulations a general point set S allows. In [12], the number of minimum pseudo-triangu... |

28 |
Kinetic maintenance of context-sensitive hierarchical representations for disjoint simple polygons
- Kirkpatrick, Speckmann
- 2002
(Show Context)
Citation Context ...ations, also called geodesic triangulations, have received considerable attention in the last few years due to their applications to visibility [10, 11], ray shooting [6], kinetic collision detection =-=[1, 8, 9]-=-, rigidity [15], and guarding [14]. While some of their interesting geometric and combinatorial properties have already been discovered [15, 13, 7, 5], many elementary open questions still remain. For... |

28 |
Minimal tangent visibility graphs
- Pocchiola, Vegter
- 1996
(Show Context)
Citation Context ...raphs where each vertex is pointed [15]. Pseudo-triangulations, also called geodesic triangulations, have received considerable attention in the last few years due to their applications to visibility =-=[10, 11]-=-, ray shooting [6], kinetic collision detection [1, 8, 9], rigidity [15], and guarding [14]. While some of their interesting geometric and combinatorial properties have already been discovered [15, 13... |

22 | Counting Triangulations and PseudoTriangulations of Wheels
- Randall, Rote, et al.
- 2001
(Show Context)
Citation Context ...ave already been discovered [15, 13, 7, 5], many elementary open questions still remain. For example, little is known about the number of minimum pseudotriangulations a general point set S allows. In =-=[12]-=-, the number of minimum pseudo-triangulations is determined for sets of points with exactly one interior point. Also, a (coarse) upper bound on the number of minimum pseudo-triangulations for sets wit... |

16 | Counting and enumerating pseudotriangulations with the greedy flip algorithm
- BRONNIMANN, KETTNER, et al.
- 2005
(Show Context)
Citation Context ...10, 11], ray shooting [6], kinetic collision detection [1, 8, 9], rigidity [15], and guarding [14]. While some of their interesting geometric and combinatorial properties have already been discovered =-=[15, 13, 7, 5]-=-, many elementary open questions still remain. For example, little is known about the number of minimum pseudotriangulations a general point set S allows. In [12], the number of minimum pseudo-triangu... |

8 | On the number of triangulations every planar point set must have
- Aichholzer, Hurtado, et al.
- 2001
(Show Context)
Citation Context ...lations as well; a simple#m n ) bound would have resulted. Figure 5 illustrates the problem in this case. Beating the threshold 2 n for triangulations is by no means trivial; the currently best bound =-=[4]-=- is # n+# ) for a small constant # > 0. Figure 5: Di#erent triangulations for S \ {p} yield the same triangulation for S when flipping towards the dashed edge. 4.1 Parity property The edge flipping op... |

1 |
Vertex #-guards in simple polygons
- Speckmann, Toth
- 2002
(Show Context)
Citation Context ...ions, have received considerable attention in the last few years due to their applications to visibility [10, 11], ray shooting [6], kinetic collision detection [1, 8, 9], rigidity [15], and guarding =-=[14]-=-. While some of their interesting geometric and combinatorial properties have already been discovered [15, 13, 7, 5], many elementary open questions still remain. For example, little is known about th... |