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Small Strictly Convex Quadrilateral Meshes of Point Sets
, 2004
"... In this paper we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3⌊n/2 ⌋ internal Steiner points are always sufficient for a convex quadrilateral mesh of n points in the plane. ..."
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Cited by 3 (0 self)
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In this paper we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3⌊n/2 ⌋ internal Steiner points are always sufficient for a convex quadrilateral mesh of n points in the plane. Furthermore, for any given n ≥ 4, there are point sets for which ⌈(n − 3)/2⌉−1 Steiner points are necessary for a convex quadrilateral mesh.
Constructing a Diffeomorphism Between a Trimmed Domain and the Unit Square
, 2003
"... This document has two objectives: decomposition of a given trimmed surface into several foursided subregions and creation of a diffeomorphism from the unit square onto each subregion. We aim at having a diffeomorphism which is easy and fast to evaluate. Throughout this paper one of our objectives i ..."
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Cited by 2 (2 self)
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This document has two objectives: decomposition of a given trimmed surface into several foursided subregions and creation of a diffeomorphism from the unit square onto each subregion. We aim at having a diffeomorphism which is easy and fast to evaluate. Throughout this paper one of our objectives is to keep the shape of the curves delineating the boundaries of the trimmed surfaces unchanged. The approach that is used invokes the use of transfinite interpolations. We will describe an automatic manner to specify internal cubic Bézierspline curves that are to be subsequently interpolated by a Gordon patch. Some theoretical criterion...
CATALOGBASED REPRESENTATION OF 2D TRIANGULATIONS
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS
, 2011
"... Several Representations and Coding schemes have been proposed to represent efficiently 2D triangulations. In this paper we propose a new practical approach to reduce the main memory space needed to represent an arbitrary triangulation, while maintaining constant time for some basic queries. This wor ..."
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Cited by 2 (2 self)
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Several Representations and Coding schemes have been proposed to represent efficiently 2D triangulations. In this paper we propose a new practical approach to reduce the main memory space needed to represent an arbitrary triangulation, while maintaining constant time for some basic queries. This work focuses on the connectivity information of the triangulation, rather than the geometric information (vertex coordinates), since the combinatorial data represents the main part of the storage. The main idea is to gather triangles into patches, to reduce the number of pointers by eliminating the internal pointers in the patches and reducing the multiple references to vertices. To accomplish this, we define and use stable catalogs of patches that are closed under basic standard update operations such as insertion and deletion of vertices, and edge flips. We present some bounds and results concerning special catalogs, and some experimental results that exhibits the practical gain of such methods.
Experimental Results on Quadrangulations of Sets of Fixed Points
"... We consider the problem of obtaining "nice" quadrangulations of planar sets of points. For many applications "nice" means that the quadrilaterals obtained are convex if possible and as"fat " or squarish as possible. For a given set of points a quadrangulation, i ..."
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Cited by 1 (0 self)
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We consider the problem of obtaining "nice" quadrangulations of planar sets of points. For many applications "nice" means that the quadrilaterals obtained are convex if possible and as&quot;fat &quot; or squarish as possible. For a given set of points a quadrangulation, if it exists, may not admit all its quadrilaterals to be convex. In such cases we desire that the quadrangulationshave as many convex quadrangles as possible. Solving this problem optimally is not practical. Therefore we propose and experimentally investigate a heuristic approach to solve this problem by converting &quot;nice &quot; triangulations to the desired quadrangulations with the aid of maximummatchings computed on the dual graph of the triangulations. We report experiments on several versions of this approach and provide theoretical justification for the good results obtained with one of these methods. The results of our experiments are particularly relevant for thoseapplications in scattered data interpolation which require quadrangulations that should stay faithful to the original data.
Which point sets admit a kangulation?
 JOURNAL OF COMPUTATIONAL GEOMETRY
, 2014
"... For k ≥ 3, a kangulation is a 2connected plane graph in which every internal face is a kgon. We say that a point set P admits a plane graph G if there is a straightline drawing of G that maps V (G) onto P and has the same facial cycles and outer face as G. We investigate the conditions under wh ..."
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For k ≥ 3, a kangulation is a 2connected plane graph in which every internal face is a kgon. We say that a point set P admits a plane graph G if there is a straightline drawing of G that maps V (G) onto P and has the same facial cycles and outer face as G. We investigate the conditions under which a point set P admits a kangulation and find that, for sets containing at least 2k2 points, the only obstructions are those that follow from Euler’s formula.
2D Triangulation Representation Using Stable Catalogs. ∗
"... The problem of representing triangulations has been widely studied to obtain convenient encodings and space efficient data structures. In this paper we propose a new practical approach to reduce the amount of space needed to represent in main memory an arbitrary triangulation, while maintaining cons ..."
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The problem of representing triangulations has been widely studied to obtain convenient encodings and space efficient data structures. In this paper we propose a new practical approach to reduce the amount of space needed to represent in main memory an arbitrary triangulation, while maintaining constant time for some basic queries. This work focuses on the connectivity information of the triangulation, rather than the geometry information (vertex coordinates), since the combinatorial data represents the main storage part of the structure. The main idea is to gather triangles into patches, to reduce the number of pointers by eliminating the internal pointers in the patches and reducing the multiple references to vertices. To accomplish this, we define stable catalogs of patches that are close under basic standard update operations such as insertion and deletion of vertices, and edge flips. We present some bounds and results concerning special catalogs, and some experimental results for the quadrilateraltriangle catalog. 1
Bichromatic Quadrangulations with Steiner Points
"... Let P be a k colored point set in general position, k ≥ 2. A family of quadrilaterals with disjoint interiors Q1,..., Qm is called a quadrangulation of P if V (Q1) ∪...∪V (Qm) = P, the edges of all Qi join points with different colors, and Q1 ∪... ∪ Qm = Conv(P). In general it is easy to see that ..."
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Let P be a k colored point set in general position, k ≥ 2. A family of quadrilaterals with disjoint interiors Q1,..., Qm is called a quadrangulation of P if V (Q1) ∪...∪V (Qm) = P, the edges of all Qi join points with different colors, and Q1 ∪... ∪ Qm = Conv(P). In general it is easy to see that not all kcolored point sets admit a quadrangulation; when they do, we call them quadrangulatable. For a point set to be quadrangulatable it must satisfy that its convex hull Conv(P) has an even number of points and that consecutive vertices of Conv(P) receive different colors. This will be assumed from now on. In this paper we study the following type of questions: Let P be a kcolored point set. How many Steiner points in the interior of Conv(P) do we need to add to P to make it quadrangulatable? When k = 2, we usually call P a bichromatic point set, and its color classes are usually denoted by R and B, i.e. the red and blue elements of P. In this paper we prove that any bichromatic � point � set P = R ∪B where R  = B  = n can be n−1 made quadrangulatable by adding at most 3 + �n � m 2 + 1 Steiner points and that 3 Steiner points are occasionally necessary. To prove the latter, we also show that the convex hull of any monochromatic point set P of n elements can be always partitioned into a set S = {S1,... � �, St} n−1 of starshaped polygons with disjoint interiors, where V (S1)∪ · · ·∪V (St) = P, and t ≤ 3 +1. For n = 3k this bound is tight. Finally we prove that there are 3colored point sets that cannot be completed to 3quadrangulatable point sets.
Upper and Lower Bounds for Strictly Convex Quadrilateral Meshes of Point Sets
, 2001
"... In this paper, we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3b 2 c internal Steiner points are always sucient for a convex quadrangulation of n points in the plane. ..."
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In this paper, we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3b 2 c internal Steiner points are always sucient for a convex quadrangulation of n points in the plane. Furthermore, for any given n 4, there are point sets for which d e 1 Steiner points may be necessary for a convex quadrangulation.
Experimental Results on Quadrangulations of Sets of Points
"... We consider the problem of obtaining "nice" quadrangulations of planar sets of points. For many applications "nice" means that the quadrilaterals obtained are convex if possible and as "fat" or as squarish as possible. For a given set of points a quadrangulation, if i ..."
Abstract
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We consider the problem of obtaining "nice" quadrangulations of planar sets of points. For many applications "nice" means that the quadrilaterals obtained are convex if possible and as "fat" or as squarish as possible. For a given set of points a quadrangulation, if it exists, may not admit all its quadrilaterals to be convex. In such cases we desire that the quadrangulations have as many convex quadrangles as possible. Solving this problem optimally is not practical. Therefore we propose and experimentally investigate a heuristic approach to solve this problem by converting "nice" triangulations to the desired quadrangulations with the aid of maximum matchings computed on the dual graph of the triangulations. We report experiments on several versions of this approach and provide theoretical justification for the good results obtained with one of these methods.