J. Bose and G. Toussaint. No quadrangulation is extremely odd. Technical Report 95-03, Department of Computer Science, University of British Columbia, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....4V I Gamma 4 = 2N triangles. Remark 9. It follows immediately from Theorem 18 that if is a four directional mesh or a type 2 partition (see e.g. Schumaker [15] then the space S 6 ( has full approximation order. Remark 10. Algorithms for quadrangulating a given data set are discussed in [5]. As shown there, it is always possible to quadrangulate a set of vertices V to create a quadrangulation of a set Omega containing the points V and whose boundary is a polygon containing an even number of points of V. In practice we will usually take Omega to be the convex hull of V. If this ....

P. Bose and G. Toussaint, No quadrangulation is extremely odd, manuscript.

....is somewhat different for quadrangulations. Indeed, it is easy to see that Omega admits a quadrangulation if and only if the number of vertices n on the boundary of Omega is even. Some algorithms for constructing quadrangulations associated with a given set of vertices have been discussed in [5], although they are not guaranteed to produce nondegenerate convex quadrangulations, even if Omega is convex. We now present two simple methods for creating nondegenerate convex quadrangulations based on subdividing a given triangulation. Method 1. Suppose 4 is a triangulation of a ....

P. Bose and G. Tousaint, No quadrangulation is extremely odd, in Algorithms and Computations, Lecture Notes in Comput. Sci., Springer Verlag, Berlin, 1004(1995), pp. 372--381. splines on triangulated quadrangulations 159

....on the given data is an interesting question. P. Bose and G. Toussaint showed that if a set of given scattered data has even number of boundary vertices, then it admits a quadrangulation which might include nonconvex quadrilaterals and described two algorithms to find such a quadrangulation (cf. [5]) Splines on FVS triangulations 7 Acknowledgments. Supported by the National Science Foundation under grant DMS 9303121. ....

Bose, P. and G. Toussaint, No quadrangulation is extremely odd, in manuscript, 1994.

....0.00 50.00 100.00 150.00 Figure 14: Distribution of angles for 100 points. Distribution of Angles (10K Pts) onion insertion Delaunay percent angle 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 0.00 50.00 100.00 150.00 Figure 15: Distribution of angles for 10K points. Bose and Toussaint [4] have recently studied a set of problems involving quadrangulation of point sets, and have obtained several interesting results. A quadrangulation of a point set S is a decomposition of the convex hull into quadrilaterals, such that each point of S is a vertex of some quadrilateral. In ....

....of S, and pair up the triangles along the path, obtaining a quadrangulation. ut In related work, Ramaswami, Ramos, and Toussaint [21] have recently studied the problem of converting triangulations to quadrangulations, e.g. by adding a small number of Steiner points. We refer the reader to [4, 21] for a wealth of other related results regarding the problems of quadrangulation and their applications. Finally, we make a remark about an alternative specification of the insertion method of obtaining Hamiltonian triangulations. The (deterministic) algorithm of Avis and ElGindy [1] does not lead ....

[Article contains additional citation context not shown here]

J. Bose and G. Toussaint. No quadrangulation is extremely odd. Technical Report 95-03, Dept. of Computer Sci., Univ. of British Columbia, 1995.

....weight triangulation for a convex polygon. Conversely, relatively little published work has been done with regards to convex quadrangulations of planar point sets. There have been some heuristic algorithms for generating well shaped quadrangulations, notably the work of Bose and Toussaint [2] using algorithms based on a serpentine triangulation of the point set. At the 4th International Workshop on Algorithms and Data Structures, WADS 95, Toussaint posed the problem of deciding whether one can decide if a set of points admits a quadrangulation where every quadrangle is convex , ....

....by the following lemma. Lemma 4.1 Let P denote a simple polygon. Also, let S denote a given point set in the plane, including the vertices of P . If P admits a quadrangulation, then ffiP must have an even number of vertices. Proof. Same type of argument as in the proof for Lemma 3. 1 in [2]. 2 Definition 4.11 Define a convex partition with respect to a point set S to be a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is a simple convex polygon. Lemma 4.2 ....

Prosenjit Bose and Godfried Toussaint. No quadrangulation is extremely odd. In Proc. 6th Annu. Internat. Sympos. Algorithms Comput. (ISAAC 95), volume 1004 of Lecture Notes in Computer Science, pages 372--381. Springer-Verlag, 1995.

..... Bhattacharya and Rosenfeld [4] have studied geometric and topological properties of ribbons. The Hamiltonian triangulation problem can be considered that of identifying if a set of points or a polygon has a triangulation that consists of a single strip (triangular ribbon) Bose and Toussaint [5] have recently studied a set of problems involving quadrangulation of point sets, and have obtained several interesting results. A quadrangulation of a point set S is a decomposition of the convex hull into quadrilaterals, such that each point of S is a vertex of some quadrilateral. In ....

J. Bose and G. Toussaint. No quadrangulation is extremely odd. Technical Report 95-03, Department of Computer Science, University of British Columbia, 1995.

..... Bhattacharya and Rosenfeld [4] have studied geometric and topological properties of ribbons. The Hamiltonian triangulation problem can be considered that of identifying if a set of points or a polygon has a triangulation that consists of a single strip (triangular ribbon) Bose and Toussaint [5] have recently studied a set of problems involving quadrangulation of point sets, and have obtained several interesting results. A quadrangulation of a point set S is a decomposition of the convex hull into quadrilaterals, such that each point of S is a vertex of some quadrilateral. In ....

J. Bose and G. Toussaint. No quadrangulation is extremely odd. Technical Report 95-03, Department of Computer Science, University of British Columbia, 1995.

....is somewhat different for quadrangulations. Indeed, it is easy to see that Omega admits a quadrangulation if and only if the number of vertices n on the boundary of Omega is even. Some algorithms for constructing quadrangulations associated with a given set of vertices have been discussed in [5], although they are not guaranteed to produce nondegenerate convex quadrangulations, even if Omega is convex. We now present two simple methods for creating nondegenerate convex quadrangulations based on subdividing a given triangulation. Fig. 3. Use of Method 1 to construct a quadrangulation ....

Bose, P. and G. Tousaint, No quadrangulation is extremely odd, in Algorithms and Computations, Lecture Notes in Comput. Sci., Springer Verlag, Berlin 1004 (1995), 372--381.

....minimum weight triangulation for a convex polygon. Conversely, relatively little published work has been done with regards to convex quadrangulations of planar point sets. There have been some heuristic algorithms for generating well shaped quadrangulations, e.g. the work of Bose and Toussaint [2] using algorithms based on a serpentine triangulation of the point set. At the 4th International Workshop on Algorithms and Data Structures, WADS 95, Toussaint posed the problem of deciding whether one can decide if a set of points admits a quadrangulation where every quadrangle is convex , ....

....is explained by the following lemma. Lemma 2.1 Let P be a simple polygon. Also, let S be a given point set in the plane, including the vertices of P . If P admits a quadrangulation, then ffiP must have an even number of vertices. Proof. Same type of argument as in the proof for Lemma 3. 1 in [2]. 2 Lemma 2.2 For any convex quadrangulation with respect to a point set, each interior vertex y is incident on an edge (x; y) that is l.w.r. y. Proof. Simple extension of Anagnostou and Corneil [1] s proof for triangulations based on the fact that in a convex quadrangulation there are no interior ....

Prosenjit Bose and Godfried Toussaint. No quadrangulation is extremely odd. In Proc. 6th Annu. Internat. Sympos. Algorithms Comput. (ISAAC 95), volume 1004 of Lecture Notes in Computer Science, pages 372--381. Springer-Verlag, 1995.

....I Gamma 4 = 2N triangles. Remark 9. It follows immediately from Theorem 18 that if is a four directional mesh or a type 2 partition (see e.g. Schumaker [15] then the space S 2 6 ( has full approximation order. Remark 10. Algorithms for quadrangulating a given data set are discussed in [5]. As shown there, it is always possible to quadrangulate a set of vertices V to create a quadrangulation of a set Omega containing the points V and whose boundary is a polygon containing an even number of points of V. In practice we will usually take Omega to be the convex hull of V. If this ....

P. Bose and G. Toussaint, No quadrangulation is extremely odd, manuscript.

....returned for the minimum weight value. Conversely, relatively little published work has been done with regards to optimized convex quadrangulations of planar point sets. There have been some heuristic algorithms for generating well shaped quadrangulations, notably the work of Bose and Toussaint [2] using algorithms based on a serpentine triangulation of the point set. At the 4th International Workshop on Algorithms and Data Structures, WADS 95, Toussaint posed the problem of deciding whether one can decide if a set of points admits a quad1 rangulation where every quadrangle is convex , ....

Prosenjit Bose and Godfried Toussaint. No quadrangulation is extremely odd. In Proc. 6th Annu. Internat. Sympos. Algorithms Comput. (ISAAC 95), volume 1004 of Lecture Notes in Computer Science, pages 372--381. Springer-Verlag, 1995.

....interpolatory criteria such as curvature and other derivative information. See [19] for a discussion of such issues. The characterization of quadrangulations of sets of objects and the design of algorithms for their efficient computation using a small number of Steiner points have only just begun [6, 9, 10, 28, 29, 30, 36]. A set of points admits a quadrangulation without Steiner points if and only if the number of points on the convex hull is even. This was first shown in [39] and may have been known by Euler [12] it was later rediscovered in [9, 10] In [36] the authors show that there are simple polygons that ....

....number of Steiner points have only just begun [6, 9, 10, 28, 29, 30, 36] A set of points admits a quadrangulation without Steiner points if and only if the number of points on the convex hull is even. This was first shown in [39] and may have been known by Euler [12] it was later rediscovered in [9, 10]) In [36] the authors show that there are simple polygons that may require n) Steiner points to be quadrangulated, and give efficient algorithms for quadrangulating an arbitrary simple polygon with a bounded number of Steiner points. In [28, 29, 30] the authors give complexity results and ....

[Article contains additional citation context not shown here]

P. Bose and G. T. Toussaint. No quadrangulation is extremely odd. In Proc. of the International Symposium on Algorithms and Computation, pages 372--281, Cairns, Australia, December 4-6 1995.

....simple polygons or polygons with points inside) admit a quadrangulation. For a survey on this topic see [31] The characterization of quadrangulations of sets of objects and the design of algorithms for their efficient computation using the minimum number of Steiner points have only just begun [6, 9, 10, 22, 23, 24, 29]. In [9, 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 [29] the authors show that there are simple polygons that may require Omega Gamma n) Steiner points to be quadrangulated, and give ....

....with points inside) admit a quadrangulation. For a survey on this topic see [31] The characterization of quadrangulations of sets of objects and the design of algorithms for their efficient computation using the minimum number of Steiner points have only just begun [6, 9, 10, 22, 23, 24, 29] In [9, 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 [29] the authors show that there are simple polygons that may require Omega Gamma n) Steiner points to be quadrangulated, and give efficient ....

[Article contains additional citation context not shown here]

P. Bose and G. T. Toussaint. No quadrangulation is extremely odd. In Proc. of the International Symposium on Algorithms and Computation, pages 372--281, Cairns, Australia, December 4-6 1995.

....(i.e. no extra points called Steiner points are permitted) then not all sets of points admit 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 typical input consists of a set of points lying in the interior of a polygon with holes [19, 22] ....

....no extra points called Steiner points are permitted) then not all sets of points admit 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 typical input consists of a set of points lying in the interior of a polygon with holes [19, 22] Since ....

[Article contains additional citation context not shown here]

P. Bose and G. T. Toussaint. No quadrangulation is extremely odd. In Proc. of the International Symposium on Algorithms and Computation, Cairns, Australia, December 4-6 1995.

No context found.

J. Bose and G. Toussaint. No quadrangulation is extremely odd. Technical Report 95-03, Department of Computer Science, University of British Columbia, 1995.

No context found.

Bose, P., and G. Toussaint, "No Quadrangulation is Extremely Odd," technical report #95-03, Dept. of Computer Science, University of British Columbia, January 1995.

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