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Convex Polygons in Geometric Triangulations
[chapter]

2015
*
Lecture Notes in Computer Science
*

We show that the maximum number

doi:10.1007/978-3-319-21840-3_24
fatcat:233glikkrnh4nhlv6nb774bn7m
*of**convex**polygons*in a*triangulation**of*n points in the plane is O(1.5029 n ). ... Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number*of**convex**polygons*in G. ... Note that f k (a, b, c)*counts*the number*of**convex**polygons*in T with k vertices, leftmost vertex a, and containing a counterclockwise*convex*arc (b, c, a), hence each*convex**polygon*is*counted*exactly ...##
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Convex Polygons in Geometric Triangulations

2017
*
Combinatorics, probability & computing
*

We show that the maximum number

doi:10.1017/s0963548317000141
fatcat:4yfjh3zprjaknf2mlhxlmzuwiu
*of**convex**polygons*in a*triangulation**of*n points in the plane is O(1.5029 n ). ... Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number*of**convex**polygons*in G. ... The authors are grateful to an anonymous reviewer for a very careful reading*of*the manuscript and for pertinent remarks. ...##
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Triangulations, visibility graph and reflex vertices of a simple polygon

1996
*
Computational geometry
*

In this paper tight lower and upper bounds for the number

doi:10.1016/0925-7721(95)00027-5
fatcat:t5smulcdyrgd3hq4wxkmdthmbm
*of**triangulations**of*a simple*polygon*are obtained as a function*of*the number*of*reflex vertices, thus relating these two shape descriptors. ... Tight bounds for the size*of*the visibility graph*of*a*polygon*are obtained too, with the same parameter. ... Lemma 9 . 9 Among all*almost*-*convex**polygons*with n vertices, k*of*them being rej? ...##
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Geometric Intersection
[chapter]

2004
*
Handbook of Discrete and Computational Geometry, Second Edition
*

the intersection

doi:10.1201/9781420035315.ch38
fatcat:k3gkhwpsprdbtijs4k46y57lqu
*of*two*convex**polygons*. ... Each*polygon*is preprocessed by computing a geodesic*triangulation**of*its exterior. ...##
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Page 5399 of Mathematical Reviews Vol. , Issue 94i
[page]

1994
*
Mathematical Reviews
*

O(n>)] for two [resp. three]

*convex**polygons**of*constant sizes moving in a nonconvex*polygon**of*size n.” ... Two examples without possibility*of*double contacts are given, one with three*polygons*(not*convex*) moving in a*polygonal*environ- ment, and one with four*convex**polygons*moving in a rectangle. ...##
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Page 444 of Mathematical Reviews Vol. , Issue 91A
[page]

1991
*
Mathematical Reviews
*

together in an orthogonally

*convex*covering*polygon*. ... It is shown that the visibility graph*of*a horizontally or vertically*convex**polygon*is a permutation graph. ...##
###
On Finding Large Polygonal Voids Using Delaunay Triangulation: The Case of Planar Point Sets
[chapter]

2014
*
Proceedings of the 22nd International Meshing Roundtable
*

This paper proposes a new method to find voids that starting from local longest-edges in a Delaunay

doi:10.1007/978-3-319-02335-9_16
dblp:conf/imr/HerviasHCF13
fatcat:pfmqi4igrne3ncl7k4gkprnd6m
*triangulation*builds the largest possible empty or*almost*empty*polygons*around them. ... A*polygon*is considered a void if its area is larger than a threshold value. The algorithm is validated in 2D points with artificially generated circular and non-*convex**polygon*voids. ... A*convex*hole or an empty*convex**polygon*defined by vertices*of*P is a*convex**polygon*that contains no point*of*P in its interior. ...##
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Counting triangulations of some classes of subdivided convex polygons
[article]

2016
*
arXiv
*
pre-print

We compute the number

arXiv:1604.02870v1
fatcat:5cczmtfkjbfpjgds5pkjlkukx4
*of**triangulations**of*a*convex*k-gon each*of*whose sides is subdivided by r-1 points. ... We connect these results with the question*of*finding the planar set*of*points in general position that has the minimum possible number*of**triangulations*- a well-known open problem from computational ... Hurtado and Noy [11] considered*triangulations**of**almost**convex**polygons*, which turn out to be equivalent to subdivided*convex**polygons*according to our terminology. ...##
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Pre-Triangulations and Liftable Complexes

2007
*
Discrete & Computational Geometry
*

complexes, and as graphs

doi:10.1007/s00454-007-9032-z
fatcat:kbnzhxg4bjhatdvey5hmerdn7y
*of*maximal locally*convex*functions. ... Keywords Pre-*triangulations*· Pseudo-*triangulations*· Liftable complexes Introduction*Polygonal*complexes in the plane have been objects*of*interest in combinatorial geometry from various points*of*view ... Observe that the*convex*hull*of*a*polygonal*region R is a*convex**polygon*whose vertices are corners*of*R. This implies that every*polygonal*region has at least 3 corners. ...##
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Survey of two-dimensional acute triangulations

2013
*
Discrete Mathematics
*

*Triangulations*are particular instances

*of*dissections (for which the above intersection condition reduces to requiring only empty interior, and any kind

*of*

*polygons*can be considered instead

*of*triangles ... Concerning the further organization

*of*this survey: the first part deals mainly with acute

*triangulations*

*of*

*polygons*, covering existence, asymptotic upper bounds, mesh generation algorithms, concrete ... An

*almost*-

*triangulation*with five or more vertices admits a non-obtuse straight-line embedding if and only if Surfaces Platonic surfaces A

*convex*surface is the boundary

*of*a compact

*convex*set in ...

##
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Fast triangulation of the plane with respect to simple polygons

1985
*
Information and Control
*

Given a

doi:10.1016/s0019-9958(85)80044-9
fatcat:wmgkkj3iv5hzdlj45jphdihbfi
*triangulation**of*the plane with respect to a set*of*k pairwise non-intersecting simple*polygons*, then the intersection*of*this set with a*convex**polygon*Q can be computed in time linear with respect ... Such a result had only be known for two*convex**polygons*. The other application improves the bound on the number*of**convex*parts into which a*polygon*can be decomposed. ... Intersection*of*a Set*of*k*Polygons*and a*Convex**Polygon*Q Shamos (1975) showed how to compute the intersection*of*two*convex**polygons*in linear time. We extend his result as follows.THEOREM 5. ...##
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Plane Geometric Graph Augmentation: A Generic Perspective
[chapter]

2012
*
Thirty Essays on Geometric Graph Theory
*

The geometric constraints on the possible new edges render some

doi:10.1007/978-1-4614-0110-0_17
fatcat:vpjhqxc6qffe3h2tjjwzle6stq
*of*the simplest augmentation problems intractable, and in many cases only extremal results are known. ... In the double zig-zag chain, P and Q are the most basic type*of**almost*-*convex**polygons*, a class introduced by Hurtado and Noy [63] . ... Using other*almost*-*convex**polygons*in a similar manner, Dumitrescu et al. [36] have very recently constructed n-element point sets that admit Ω(8.65 n )*triangulations*. ...##
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Resolving Loads with Positive Interior Stresses
[chapter]

2009
*
Lecture Notes in Computer Science
*

As an application, we discuss how to compute the maximal locally

doi:10.1007/978-3-642-03367-4_46
fatcat:n7l6e6tr4vevxiq4yxhtzrzibe
*convex*function for a*polygon*whose corners lie on its*convex*hull. ... For the case where the forces appear only at*convex*hull vertices we show that the pseudo-*triangulation*that resolves the load can be computed as weighted Delaunay*triangulation*. ... A pseudo-*triangulation**of*P is a partition*of*the*convex*hull*of*P into*polygons*with three corners. such that every p i is part*of*some*polygon*. ...##
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Pseudo-Triangulations - a Survey
[article]

2007
*
arXiv
*
pre-print

A pseudo-triangle is a simple

arXiv:math/0612672v2
fatcat:adhyppd3wjbhnfmdky5dbxjxzu
*polygon*with three*convex*vertices, and a pseudo-*triangulation*is a face-to-face tiling*of*a planar region into pseudo-triangles. ... Pseudo-*triangulations*appear as data structures in computational geometry, as planar bar-and-joint frameworks in rigidity theory and as projections*of*locally*convex*surfaces. ... This is in several ways analogous to the ubiquitous*triangulations*which appear*almost*everywhere in Combinatorial Geometry, and has led to the investigation*of*similar questions:*counting*, enumeration ...##
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Counting Carambolas

2015
*
Graphs and Combinatorics
*

Configurations

doi:10.1007/s00373-015-1621-7
fatcat:cf76frghsvgilmfrcrkxl3rb2q
*of*interest include*convex**polygons*, star-shaped*polygons*and monotone paths. We also consider related problems for directed planar straight-line graphs. ... We give upper and lower bounds on the maximum and minimum number*of*geometric configurations*of*various kinds present (as subgraphs) in a*triangulation**of*n points in the plane. ...*Convex**polygons*. Every n-vertex*triangulation*has Θ(n)*convex*faces, hence Ω(n) is a natural lower bound for the number*of**convex**polygons*. ...
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