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### Convex Polygons in Geometric Triangulations [chapter]

Adrian Dumitrescu, Csaba D. Tóth
2015 Lecture Notes in Computer Science
We show that the maximum number 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  ...

### Convex Polygons in Geometric Triangulations

ADRIAN DUMITRESCU, CSABA D. TÓTH
2017 Combinatorics, probability & computing
We show that the maximum number 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.  ...

### Triangulations, visibility graph and reflex vertices of a simple polygon

F. Hurtado, M. Noy
1996 Computational geometry
In this paper tight lower and upper bounds for the number 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?  ...

### Geometric Intersection [chapter]

David Mount
2004 Handbook of Discrete and Computational Geometry, Second Edition
the intersection of two convex polygons.  ...  Each polygon is preprocessed by computing a geodesic triangulation of its exterior.  ...

### 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.  ...

### 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]

Carlos Hervías, Nancy Hitschfeld-Kahler, Luis E. Campusano, Giselle Font
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 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.  ...

### Counting triangulations of some classes of subdivided convex polygons [article]

Andrei Asinowski, Christian Krattenthaler, Toufik Mansour
2016 arXiv   pre-print
We compute the number 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  considered triangulations of almost convex polygons, which turn out to be equivalent to subdivided convex polygons according to our terminology.  ...

### Pre-Triangulations and Liftable Complexes

Oswin Aichholzer, Franz Aurenhammer, Thomas Hackl
2007 Discrete & Computational Geometry
complexes, and as graphs 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.  ...

### Survey of two-dimensional acute triangulations

Carol T. Zamfirescu
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  ...

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

Stefan Hertel, Kurt Mehlhorn
1985 Information and Control
Given a 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.  ...

### Plane Geometric Graph Augmentation: A Generic Perspective [chapter]

Ferran Hurtado, Csaba D. Tóth
2012 Thirty Essays on Geometric Graph Theory
The geometric constraints on the possible new edges render some 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  .  ...  Using other almost-convex polygons in a similar manner, Dumitrescu et al.  have very recently constructed n-element point sets that admit Ω(8.65 n ) triangulations.  ...

### Resolving Loads with Positive Interior Stresses [chapter]

Günter Rote, André Schulz
2009 Lecture Notes in Computer Science
As an application, we discuss how to compute the maximal locally 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.  ...

### Pseudo-Triangulations - a Survey [article]

Guenter Rote, Francisco Santos, Ileana Streinu
2007 arXiv   pre-print
A pseudo-triangle is a simple 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  ...

### Counting Carambolas

Adrian Dumitrescu, Maarten Löffler, André Schulz, Csaba D. Tóth
2015 Graphs and Combinatorics
Configurations 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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