## Converting triangulations to quadrangulations (1983)

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Venue: | PHYS |

Citations: | 24 - 7 self |

### BibTeX

@ARTICLE{Ramaswami83convertingtriangulations,

author = {Suneeta Ramaswami and Pedro Ramos and Godfried Toussaint},

title = {Converting triangulations to quadrangulations},

journal = {PHYS},

year = {1983},

volume = {49},

pages = {413}

}

### Years of Citing Articles

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### Citations

1540 |
Network Flows: Theory, Algorithms, and Applications
- Ahuja, Magnanti, et al.
- 1993
(Show Context)
Citation Context .... A perfect matching is a matching such that every node in V belongs to an edge of the matching. Note that this is slightly different from some definitions of perfect matching found in the literature =-=[1, 30], where th-=-e definition allows one extra "free" i.e. unmatched node (when jV j is odd, there will be at least one unmatched node in a maximum matching). Our definition is more appropriate in the contex... |

1356 | Combinatorial Optimization: Algorithms and Complexity - Papadimitriou, Steiglitz - 1982 |

296 |
Triangulating a simple polygon in linear time
- Chazelle
- 1991
(Show Context)
Citation Context ...To find an appropriate placement of the Steiner point, we may triangulate the simple polygon (or polygons) that lie outside P and within the convex hull of P , in O(n) time using Chazelle's algorithm =-=[12]-=-. The Steiner point for e can be placed anywhere inside the triangle incident on e (and in the exterior of P ). If e is an edge of the convex hull, then the Steiner point can be located in the interio... |

283 |
Introduction to Geometry
- Coxeter
- 1969
(Show Context)
Citation Context ...gon lateral." All polygons except (for some unknown reason) the polygon of four vertices, are referred to by their number of vertices (angles) rather than their sides (latus). In the words of Cox=-=eter [14], "it-=- is more usual to call this a quadrilateral, but to do so is unreasonable, as the word triangle refers to its vertices rather than its sides, and so too does the word pentagon." We assume that a ... |

244 | Easy problems for tree decomposable graphs
- Arnborg, Lagergren, et al.
- 1991
(Show Context)
Citation Context ...s added structure. Recently a general theory has been developed for solving a variety of optimization problems on a class of graphs (called tree-decomposable) in linear time using dynamic programming =-=[8, 4]-=-. This class of problems includes maximum matchings and the class of graphs includes trees. On the other hand a simpler and more straight-forward linear-time algorithm for computing maximum matchings ... |

188 | Mesh Generation and Optimal Triangulation
- Bern, Eppstein
- 1992
(Show Context)
Citation Context ...inite elements are, as the name implies, triangles. As a result, triangulations of sets (such as sets of points, line segments, polygons, etc.) have been studied in depth and much is known about them =-=[6]-=-. However, in some situations for both the finite element and the scattered data interpolation problems, it is preferable that the finite elements be quadrangles (quadrilaterals) instead of triangles.... |

155 |
A combinatorial theorem in plane geometry
- Chvatal
- 1975
(Show Context)
Citation Context ...gonals between pairs of vertices. Although the Hamiltonian approach gives a marked improvement in the number of Steiner points used, we show that by using coloring arguments for triangulated polygons =-=[13, 18]-=-, we can further reduce the number of Steiner points by a factor of three and this is optimal. Before proceeding, we make our definition of Steiner points more precise. As pointed out in the introduct... |

152 | An O( p|V | · |E|) algorithm for finding maximum matchings in general graphs - Micali, Vazirani - 1980 |

87 |
Two theorems in graph theory
- Berge
- 1957
(Show Context)
Citation Context ...: this is because we can obtain a matching of size jM j + 1 by interchanging the matching and free edges along the path. Less obviously, the converse is also true and we have the following: Lemma 3.1 =-=[5, 29]-=- M is a maximum matching of a graph G = (V; E) if and only if G has no augmenting paths with respect to M . We now give the description of simple linear-time algorithms, which we call the percolation ... |

73 |
A short proof of Chvatal's watchman theorem
- Fisk
- 1978
(Show Context)
Citation Context ...gonals between pairs of vertices. Although the Hamiltonian approach gives a marked improvement in the number of Steiner points used, we show that by using coloring arguments for triangulated polygons =-=[13, 18]-=-, we can further reduce the number of Steiner points by a factor of three and this is optimal. Before proceeding, we make our definition of Steiner points more precise. As pointed out in the introduct... |

66 | Hamiltonian triangulations for fast rendering
- Arkin, Held, et al.
- 1996
(Show Context)
Citation Context ... always use 3n \Gamma 5 Steiner points in a triangulated simple n-gon. Another approach that uses about one third as many Steiner points is via the Hamiltonian triangulation algorithm of Arkin et al. =-=[3]-=-. With a very different goal in mind, namely, fast rendering in computer graphics, Arkin et al. proposed an elegant method of obtaining what they call a Hamiltonian-cycle triangulation. Such a triangu... |

59 |
Finite Element Mesh Generation Method* A Review and Classification
- Ho-Le
- 1988
(Show Context)
Citation Context ... processes, such as fluid flow in injection molding, by solving complicated systems of partial differential equations [11]. To make this task easier, the method of finite elements is usually employed =-=[20]-=-. In this approach a solid model of the object under study (or its bounding surface) is divided up into small pieces determined by data points sampled on the object's surface. In scattered bivariate d... |

47 |
Linear-Time Computation of Optimal Subgraphs of Decomposable Graphs
- Bern, Lawler, et al.
- 1987
(Show Context)
Citation Context ...s added structure. Recently a general theory has been developed for solving a variety of optimization problems on a class of graphs (called tree-decomposable) in linear time using dynamic programming =-=[8, 4]-=-. This class of problems includes maximum matchings and the class of graphs includes trees. On the other hand a simpler and more straight-forward linear-time algorithm for computing maximum matchings ... |

45 |
Traditional galleries require fewer watchmen
- Kahn, Klawe, et al.
- 1983
(Show Context)
Citation Context ...ns of orthogonal polygons are not interesting and have not been studied. An existential proof that orthogonal polygons always admit convex quadrangulations was first given by Kahn, Klawe and Kleitman =-=[23]-=-. A constructive proof with an O(n) time algorithm was first obtained by Sack & Toussaint [32] for star-shaped polygons and subsequently generalized to run in O(n log n) time for arbitrary simple orth... |

25 |
Delaunay-based representation of surfaces defined over arbitrarily shaped domains
- Floriani, Falcidieno, et al.
- 1985
(Show Context)
Citation Context ...e application of such a problem in the area of computer cartography is the construction of approximate models of terrains from data consisting of the elevation at a given finite set of sampled points =-=[15]. To facil-=-itate this process the data points in the plane are used to divide it into small pieces. Each such piece then gives rise to a surface patch and these surface patches are finally "stitched" t... |

23 |
An algorithm for a minimum cover of a graph
- Norman, Rabin
- 1959
(Show Context)
Citation Context ...: this is because we can obtain a matching of size jM j + 1 by interchanging the matching and free edges along the path. Less obviously, the converse is also true and we have the following: Lemma 3.1 =-=[5, 29]-=- M is a maximum matching of a graph G = (V; E) if and only if G has no augmenting paths with respect to M . We now give the description of simple linear-time algorithms, which we call the percolation ... |

19 |
On the dimension of spaces of piecewise polynomials in two variables
- Schumaker
- 1979
(Show Context)
Citation Context ...on the object's surface. In scattered bivariate data interpolation one is required to construct a bivariate function (or surface) that fits data that has been collected at sampled points on the plane =-=[34]-=-. One application of such a problem in the area of computer cartography is the construction of approximate models of terrains from data consisting of the elevation at a given finite set of sampled poi... |

17 | No quadrangulation is extremely odd
- Bose, Toussaint
- 1995
(Show Context)
Citation Context ... a quadrangulation. The characterization of quadrangulations of point sets and the design of algorithms for their efficient computation using the minimum number of Steiner points have only just begun =-=[10]-=-. In [10] it is shown that a set of points admits a quadrangulation without Steiner points if and only if the number of points on the convex hull is even. In practical problems faced by engineers, the... |

17 |
Decomposing Polygonal Regions into Convex Quadrilaterals
- Lubiw
(Show Context)
Citation Context ...ck & Toussaint [32] for star-shaped polygons and subsequently generalized to run in O(n log n) time for arbitrary simple orthogonal polygons by Sack [31]. Edelsbrunner, O'Rourke and Welzl [16], Lubiw =-=[27]-=- and Sack & Toussaint [33], among others, later obtained additional constructive variants with similar time complexities. An orthogonal polygon with holes does not necessarily admit a convex quadrangu... |

17 |
Automatic Mesh Generation Using the Symmetric Axis Transformation
- Srinivasan, Nackman, et al.
- 1992
(Show Context)
Citation Context ...omputing quadrangulations, whereas triangulations have been well studied for several decades [6], engineers have devoted some attention to the problem of convertingstriangulations to quadrangulations =-=[19, 22, 35]-=-. These methods however are heuristic, conceptually rather cumbersome and may require many Steiner points. For example, Johnston et al. [22] integrate several heuristics into a system that automatical... |

16 |
Quadrilateral mesh generation in polygonal regions
- Joe
- 1995
(Show Context)
Citation Context ...s by using quadrangles rather than triangles [2]. Unfortunately, not much is known about quadrangulations of point sets and good quadrangular meshes are harder to generate than good triangular meshes =-=[21]-=-. In fact, if edges are allowed to be inserted only between the given data points (i.e., no extra points called Steiner points are permitted) then not all sets of points admit a quadrangulation. The c... |

15 |
Stationing guards in rectilinear art galleries
- Edelsbrunner, O’Rourke, et al.
- 1984
(Show Context)
Citation Context ...tained by Sack & Toussaint [32] for star-shaped polygons and subsequently generalized to run in O(n log n) time for arbitrary simple orthogonal polygons by Sack [31]. Edelsbrunner, O'Rourke and Welzl =-=[16]-=-, Lubiw [27] and Sack & Toussaint [33], among others, later obtained additional constructive variants with similar time complexities. An orthogonal polygon with holes does not necessarily admit a conv... |

15 |
A Mesh Generator for Automatically Subdividing Irregular Polygons into Quadrilaterals
- Heighway
(Show Context)
Citation Context ...s if and only if the number of points on the convex hull is even. In practical problems faced by engineers, the typical input consists of a set of points lying in the interior of a polygon with holes =-=[19, 22]-=-. Since little is known about computing quadrangulations, whereas triangulations have been well studied for several decades [6], engineers have devoted some attention to the problem of convertingstria... |

13 |
A quadrilateral finite element including vertex rotations for plane elasticity analysis
- Allman
- 1988
(Show Context)
Citation Context ...r triangulations for the problem of scattered data interpolation [26] and that improvements in elasticity analysis can be obtained in finite element methods by using quadrangles rather than triangles =-=[2]-=-. Unfortunately, not much is known about quadrangulations of point sets and good quadrangular meshes are harder to generate than good triangular meshes [21]. In fact, if edges are allowed to be insert... |

13 |
Automatic conversion of triangular finite meshes to quadrilateral elements
- Johnston, Sullivan, et al.
- 1991
(Show Context)
Citation Context ...s if and only if the number of points on the convex hull is even. In practical problems faced by engineers, the typical input consists of a set of points lying in the interior of a polygon with holes =-=[19, 22]-=-. Since little is known about computing quadrangulations, whereas triangulations have been well studied for several decades [6], engineers have devoted some attention to the problem of convertingstria... |

12 |
Strictly Convex Quadrilateralizations of Polygons
- Everett, Lenhart, et al.
- 1992
(Show Context)
Citation Context ...at the quadrangles obtained be strict quadrangles, i.e., that quadrangles not contain three collinear vertices, which would in effect make them triangles. For example, in some mesh generation methods =-=[17]-=- and in the recent efficient scattered data interpolation algorithms [26], the quadrangles must be strict. Although strict quadrangulations may be obtained by adding Steiner points on the boundary or ... |

12 | Quadrangulations of planar sets
- Toussaint
- 1995
(Show Context)
Citation Context ... a convex quadrangulation and Lubiw [27] showed that to determine if this is possible is NP-complete. For references to additional special cases of quadrangulation problems, the reader is referred to =-=[36]-=-. In this paper we study the problem of converting general triangulated domains to quadrangulations, under a variety of constraints. We focus on a careful study of quadrangulating simple polygons and ... |

11 |
Injection Moulding of Plastic Components
- Bown
- 1979
(Show Context)
Citation Context ...problem in the manufacturing industry concerns the simulation of a wide variety of processes, such as fluid flow in injection molding, by solving complicated systems of partial differential equations =-=[11]-=-. To make this task easier, the method of finite elements is usually employed [20]. In this approach a solid model of the object under study (or its bounding surface) is divided up into small pieces d... |

11 |
Scattered data interpolation using piecewise polynomials of degree 6
- Lai, Schumaker
- 1997
(Show Context)
Citation Context ...angles (quadrilaterals) instead of triangles. For example, it has recently been shown that quadrangulations have several advantages over triangulations for the problem of scattered data interpolation =-=[26]-=- and that improvements in elasticity analysis can be obtained in finite element methods by using quadrangles rather than triangles [2]. Unfortunately, not much is known about quadrangulations of point... |

10 |
Three-Coloring the Vertices of a Triangulated Simple Polygon
- Kooshesh, Moret
- 1992
(Show Context)
Citation Context ...n from the triangulated polygon. To see that these Steiner points can be located in O(n) time, consider the following. The triangulated polygon can be three-colored in linear time (Kooshesh and Moret =-=[25]-=-). The edge on which a guard is placed gives us the fan-arm e outside which we place the Steiner point. To find an appropriate placement of the Steiner point, we may triangulate the simple polygon (or... |

9 |
Guard placement in rectilinear polygons
- SACK, TOUSSAINT
- 1988
(Show Context)
Citation Context ...ar-shaped polygons and subsequently generalized to run in O(n log n) time for arbitrary simple orthogonal polygons by Sack [31]. Edelsbrunner, O'Rourke and Welzl [16], Lubiw [27] and Sack & Toussaint =-=[33]-=-, among others, later obtained additional constructive variants with similar time complexities. An orthogonal polygon with holes does not necessarily admit a convex quadrangulation and Lubiw [27] show... |

8 |
Drawing the planar dual
- Bern, Gilbert
- 1992
(Show Context)
Citation Context ...n as in Figure 2 (a). First, a planar dual tree is inserted in the triangulated polygon. That this can always be done in a triangulation or convex quadrangulation was first proved by Bern and Gilbert =-=[7]-=-. Next, in each triangle the node in the dual tree corresponding to this triangle is connected with edges to the three vertices of the triangle. Finally, the original diagonals of the triangulated pol... |

6 |
An 0 (n log n) algorithm for decomposing simple rectilinear polygons into convex quadrilaterals
- Sack
- 1982
(Show Context)
Citation Context ...ith an O(n) time algorithm was first obtained by Sack & Toussaint [32] for star-shaped polygons and subsequently generalized to run in O(n log n) time for arbitrary simple orthogonal polygons by Sack =-=[31]-=-. Edelsbrunner, O'Rourke and Welzl [16], Lubiw [27] and Sack & Toussaint [33], among others, later obtained additional constructive variants with similar time complexities. An orthogonal polygon with ... |

5 |
A linear-time algorithm for decomposing rectilinear star-shaped polygons intoconvex quadrilaterals
- Sack, Toussaint
- 1981
(Show Context)
Citation Context ...that orthogonal polygons always admit convex quadrangulations was first given by Kahn, Klawe and Kleitman [23]. A constructive proof with an O(n) time algorithm was first obtained by Sack & Toussaint =-=[32]-=- for star-shaped polygons and subsequently generalized to run in O(n log n) time for arbitrary simple orthogonal polygons by Sack [31]. Edelsbrunner, O'Rourke and Welzl [16], Lubiw [27] and Sack & Tou... |

3 |
Experimental comparison of quadrangulation algorithms for sets of points
- Bose, Ramaswami, et al.
- 1996
(Show Context)
Citation Context ... theorem in graph theory (Petersen's theorem) that says that every 3-regular graph without cut edges has a perfect matching. We will not go into the specifics here, but refer the interested reader to =-=[9]-=- for details, where we present experimental results on computing quadrangulations of random sets of points by utilizing some of the ideas presented in this paper. 5 Conclusions We presented efficient ... |

3 |
den Driessche, “Linear algorithms for testing the sign stability of a matrix and for finding z-maximum matchings in acyclic graphs
- Klee, van
- 1977
(Show Context)
Citation Context ...ss of graphs includes trees. On the other hand a simpler and more straight-forward linear-time algorithm for computing maximum matchings of acyclic graphs was discovered by Klee and van den Driessche =-=[24]-=- twenty years ago, although this work seems to be unknown in graph theory circles. We can use the latter algorithm to determine if the triangulated polygon admits a quadrangulation without Steiner poi... |

2 |
An O(jV j 1=2 jEj) Algorithm for Finding Maximum Matching in General Graphs
- Micali, Vazirani
- 1980
(Show Context)
Citation Context ...aximum matching answers the question of whether the triangulation admits a quadrangulation without the use of Steiner points. Using the fastest matching algorithm available due to Micali and Vazirani =-=[28]-=- this can be accomplished in O(n 1:5 ) time. On the other hand, if the domain is a triangulated simple polygon then the dual graph is a tree and maximum matchings can be computed faster by exploiting ... |

2 | Scattered data interpolation using C 2 piecewise polynomials of degree six - Lai, Schumaker - 1994 |