M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. S. Tan. Edge insertion for optimal triangulations. Discrete Comput. Geom., 10(1):47--65, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The upper bound 3c t # max (2 sin # max ) can be loosely decomposed into a size contribution (3 2)c t # max and a shape contribution 1 sin # max . This seems to suggest that a triangular mesh generator should seek to minimize the maximum angle, and algorithms for that purpose are available [18, 8]. However, this inference demonstrates the folly of too eagerly separating element size from element shape: the triangulation that optimizes # max sin # max (namely, the Delaunay triangulation) and the triangulation that optimizes 1 sin # max are not the same, and the former is more germane to ....

Marshall Bern, Herbert Edelsbrunner, David Eppstein, Scott Mitchell, and Tiow Seng Tan. EdgeInsertion for Optimal Triangulations. Discrete & Computational Geometry 10:47--65, 1993.

....the maximum angle, or minimizing the longest edge. Edelsbrunner et al. 28] 27] gave O(n 2 log n) time and O(n 2 ) time algorithms, respectively, for computing triangulations optimal in these respects. The former algorithm is based on an edge insertion paradigm which is shown in Bern et al. [7] to lead in polynomial time to triangulations with maxmin triangle height, minmax triangle eccentricity, and minmax gradient surface, respectively. Minimum weight triangulations. Most longstanding open is another optimal triangulation problem, the minimum weight triangulation, in which the ....

M.Bern, H.Edelsbrunner, D.Eppstein, S.Mitchell, T.S.Tan: `Edge insertion for optimal triangulations', Discrete&Computational Geometry 10 (1993) pp. 47-65

....(V; E) as a subset of a geometric graph G = V; E) with E E is studied by Lloyd [9] Using a reduction from 3 SAT he showed that this triangulation problem is NP complete. Given a plane geometric graph with or without constraining edges, several optimal triangulation problems have been studied [1, 2, 3]. Optimal means that the form of the triangles or the triangulations is optimized. In contrast to polynomial algorithms in [1, 2, 3] we give the rst negative result for an optimal triangulation problem. The NP completeness of a similar problem to triangulate a planar graph while minimizing the ....

....this triangulation problem is NP complete. Given a plane geometric graph with or without constraining edges, several optimal triangulation problems have been studied [1, 2, 3] Optimal means that the form of the triangles or the triangulations is optimized. In contrast to polynomial algorithms in [1, 2, 3], we give the rst negative result for an optimal triangulation problem. The NP completeness of a similar problem to triangulate a planar graph while minimizing the maximum degree has been proved by Kant and Bodlaender [7] One di erence in [7] to our considered problem and to the studied ....

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M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell and T.S. Tan, Edge insertion for optimal triangulations, Proc. Latin America Theoretical Informat. (1992).

.... if angles of triangles become too large, the discretization error in the finite element solution is increased and, if the angles become too small, the condition number of the element matrix is increased [1, 10, 11] Polynomial time algorithms have been developed in determining those triangulations [2, 7, 8, 15]. In computational geometry another important research object is to compute the minimum weight triangulation. The weight of a triangulation is defined to be the sum of the Euclidean lengths of the edges in the triangulation. Despite the intensive study made during the lase two decades, it remains ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. S. Tan, "Edge insertion for optimal triangulations", Discrete Comput. Geom., 10, pp.47-65, 1993.

....the lower shaded disk such that bc xy. So xy belongs to the fi 1 skeleton of fb; c; x; yg but the MWT of fb; c; x; yg contains bc instead of xy. In Section 2, we shall review Keil s proof. Our result is presented in Section 3. 2 Preliminaries Keil s proof follows the edge insertion paradigm [2]. Assume to the contrary that xy is an edge of a fi skeleton that does not belong to a MWT T . The strategy is to add xy to T and remove the existing edges that intersect xy. Then the two resulting polygonal regions on both sides of xy are retriangulated carefully to obtain a new triangulation. A ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. Tan, Edge insertion for optimal triangulations, Discrete & Computational Geometry, 10 (1993), pp. 47--65.

....sin , where = tan Gamma1 (3= q 2 p 3) is indeed a lower bound on fi for fi skeleton to be a subgraph of a MWT. In Section 2, we shall review Keil s proof. Our result is presented in Section 3. b x y c a Figure 1. 2 2 Preliminaries Keil s proof follows the edge insertion paradigm [2]. Assume to the contrary that xy is an edge of a fi skeleton that does not belong to a MWT T . The strategy is to add xy to T and remove the existing edges that intersect xy. Then the two resulting polygonal regions on both sides of xy are retriangulated carefully to obtain a new triangulation. A ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. Tan, Edge insertion for optimal triangulations, Discrete & Computational Geometry, 10 (1993), pp. 47--65. 11

....information. In this paper we examine how the information provided by a set of images of the object can be used to guide selection of a surface triangulation. Much attention has been paid to the task of obtaining a surface mesh from 3D data points on an unknown surface in the graphics literature [1, 2, 3, 4, 6, 9]. Typically geometric properties of the mesh are optimized, such as relative shape, size and orientation of triangles, in the hope that this will result in faithfulness to an unknown surface. This works well for dense points on smooth surfaces, but when the features are sparse and the object ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell & T. S. Tan, Edge-insertion for optimal triangulations, Proc. Latin Amer. Sympos. Theoret. Inf., vol. 583 of Lect. Notes Comp. Sci., Springer-Verlag, Sao Paulo, 1992, 46--60.

....edge in the triangulation, removing the edges it crosses and triangulating the polygonal regions adjacent to the newly introduced edge. It can be used to compute the triangulation which minimizes the maximum angle in time O(n 2 log n) 11] There is a generalized treatment of edge insertion in [1] using the concept of anchor vertices. The idea is that it is not possible to improve a triangulation without inserting a new edge throught the anchor vertex which crosses the triangle edge adjacent to it. In an optimal triangulation, all the vertices are anchor vertices. It is shown that edge ....

M. Bern, H. Edelsbrunner, D. Eppsein, S. Mitchell, and T. S. Tan. Edge-insertion for optimal triangulations. In Proc. Latin American Theoretical Informatics, 1992.

.... if angles of triangles become too large, the discretization error in the finite element solution is increased and, if the angles become too small, the condition number of the element matrix is increased [1, 15] Polynomial time algorithms have been developed for determining those triangulations [5, 12, 13, 21]. In computational geometry computing the minimum weight triangulation is another important research topic. The weight of a triangulation is defined to be the sum of the Euclidean lengths of the edges in the triangulation. Despite the intensive study made during the lase two decades, it remains ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. S. Tan, "Edge insertion for optimal triangulations", Discrete and Computational Geometry 10 (1993) 47-65.

....with an arbitrary triangulation T and iterates until no improvement is possible. A single iteration adds a new edge to the triangulation. All edges that intersect this new edge must of course be deleted. The resulting polygons are now retriangulated. Delicate details of this scheme can be found in [2, 12]. Companion to video 3 9 The implementation supports the case where the original triangulation contains constraining edges. Furthermore, it lexicographically optimizes the entire vector of measures, not just the worst one. Because of this property it usually computes a unique optimum. For the ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchel, and T. S. Tan. Edge insertion for optimal triangulations. In Proc. 1st Latin American Sympos. Theoret. Informatics, pages 46--60, 1992.

....but for most applications one usually prefers a triangulation which is the closest to an equi angular one. One of the most common criteria is choosing the triangulation in which the smallest angle is maximal (the Min Max Angle criterion) A good survey of various min max criteria can be found in [3]. Let T be a triangulation of S. If for each triangle in T the inscribing circle contains no other point of S (except the three points that form the triangle) then T is the Delaunay Triangulation DT(S) If the above mentioned general position assumption holds, then the DT exists and it is unique. ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell and T.S. Tan, "Edge Insertion for Optimal Triangulations", 1st Latin American Symposium on Theoretical Informatics, pp. 46-60, 1992.

....n is the number of edges. Note that if all triangles are nonobtuse, the strategy of Observation 4 above is much more efficient than the one just described. To learn a polygon shaped concept using this method, one would first triangulate the polygon in such a way as to maximize h. Bern et al. [29] have developed an algorithm that can construct a triangulation of a polygon in O(n 2 log n) time that maximizes the minimum triangle altitude. Although this result does not give a lower bound for h, it does provide the maximum possible value for h. However, this method does not necessarily ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and S. Tan, "Edge insertion for optimal triangulations", in LATIN '92: 1st Latin American Symposium on Theoretical Informatics, I. Simon, Ed., Berlin, 1992, pp. 46--60, Springer-Verlag.

....theorem. Theorem 2 Let B be a box. For any triangle T on a facet of B, let r be the radius of the largest inscribed circle. Then r k Delta ff Delta h(B) where k is a constant. General results have been obtained for twodimensional triangulations with guaranteed inscribedcircle radius bounds. Bern, Edelsbrunner, Eppstein, Mitchell, and Tan [1991] have a result concerning optimal two dimensional triangulation of polygons, assuming new points cannot be introduced. Bern, Dobkin, and Eppstein [1991] have similar results in the case that new points can be introduced. We have not been able to incorporate these algorithms in the present version ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. S. Tan [1991], Edge-insertion for optimal triangulations, preprint.

....constrained triangulations that exactly optimize some measure. Edelsbrunner, Tan, and Waupotitsch [1990] introduce the edge insertion paradigm, which is a global generalization of local edge flip. Edge insertion can be used to find a constrained triangulation that minimizes the maximum angle. Bern, Edelsbrunner, Eppstein, Mitchell, and Tan [1992] show that edge insertion may also be used to find optimal constrained triangulations for any measure for which every triangle has at least one anchor vertex. This property states that for any triple of vertices of the input forming a triangle, no constrained triangulation can have measure better ....

....an approximately minimum weight triangulation. No large and no small angles are achieved simultaneously in Baker, Gross and Rafferty [1988] and in Melissaratos and Souvaine [1992] However, the cardinality of a triangulation with no small angles is doomed to be dependent on the input geometry. Bern and Eppstein [1992] summarizes much of the Steiner triangulation literature. Mitchell[1993] presents the only other known algorithm for a covering triangulation with a provable bound on triangle shape. Given a PSLG, the algorithm generates a triangulation whose minimum angle is at least a constant factor times an ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. S. Tan [1992], Edge-Insertion for Optimal Triangulations, In I. Simon, ed., LATIN '92: Proceedings of the 1st Latin American Symposium on Theoretical Informatics, Sao Paulo, Brazil. Vol 583 of Lecture Notes in Computer Science, 46-60. Springer-Verlag, Berlin.

....h 0 in the previous lemma is at least kffh(B) by Theorem 4. Moreover, t in the previous lemma was shown to be at least k Delta ff Delta h(B) in the proof of Lemma 6. General results have been obtained for two dimensional triangulations with guaranteed inscribed circle radius bounds. Bern, Edelsbrunner, Eppstein, Mitchell, and Tan [1991] have a result concerning optimal two dimensional triangulation of polygons, assuming new points cannot be introduced. Bern, Dobkin, and Eppstein [1991] have similar results in the case that new points can be introduced. We have not been able to incorporate these algorithms in the present version ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. S. Tan [1991], Edge-insertion for optimal triangulations, preprint.

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M. Bern, D. Eppstein, H. Edelsbrunner, S. Mitchell, and T.S. Tan. Edge insertion for optimal triangulations. 1st Latin American Symp. on Theoretical Informatics, 1992, 46--60.

....the paradigm and state two conditions for criteria that can be optimized. Four examples of criteria known to satisfy these conditions are the min max angle, max min height, min max slope and min max eccentricity; these are discussed in Chapter 4. The material of these two chapters also appears in [ETW90, BEEMT92]. Chapter 5 presents a quadratic time solution to the min max length triangulation problem of a point set. The result is applicable to edge lengths measured by an arbitrary normed metric, including the Euclidean distance and the more general l p metrics. It is currently the only (nontrivial) ....

....visited Xerox 49 PARC in August 1991, he solved the min max slope problem with Marshall Bern and David Eppstein (University of California, Irvine) by relaxing condition (II) to (I) Subsequently, Marshall Bern found the solution to the min max eccentricity problem. These results are collected in [BEEMT92]. Roman Waupotitsch has implemented and experimented with most of these algorithms [EdWa92, Waup92] The programs are currently available via anonymous ftp from the directory SGI MinMaxer at site ftp.ncsa.uiuc.edu . 50 Chapter 5 Minimizing the Maximum Length This chapter is devoted to the ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell and T. S. Tan. Edge insertion for optimal triangulations. In "Proc. 1st Latin American Sympos. Theoretical Inform., 1992", Lecture Notes in Comput. Sci. 583 (1992), 46--60. Also to appear in Discrete Comp. Geom.

....As mentioned in Section 5.5, edge flipping can also be used as a general optimization heuristic. For example, edge flipping works reasonably well for minimizing the maximum angle [53] but it does not in general find a global optimum. A more powerful local improvement method called edge insertion [23, 53] exactly solves the minmax angle problem, as well as several other minmax optimization problems. Edge insertion starts from an arbitrary triangulation and repeatedly inserts candidate edges. If minimizing the maximum angle is the goal, the candidate edge e subdivides the maximum angle; in general ....

.... add e and remove all edges crossed by e try to retriangulate by removing ears better than abc if retriangulation succeeds then mark bc else mark e and undo e s insertion endif endwhile Edge insertion can compute the minmax eccentricity triangulation or the minmax slope interpolating surface [23] in time O(n 3 ) By inserting candidate edges in a certain order and saving old partial triangulations, the running time can be improved to O(n 2 log n) for minmax angle [53] and maxmin triangle height. We close with some results for two other optimization criteria: maximum edge length and ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T.-S. Tan. Edge-insertion for optimal triangulations. Disc. and Comp. Geometry, 10:47--65, 1993.

....in which one optimally completes the triangulation of a convex or simple polygon can be solved by dynamic programming [78] however even this is not obvious for the minimum dilation triangulation. A solution to this subproblem might have implications in allowing the powerful edge insertion method [16, 51] to be applied to the point set version of the problem. Open Problem 7. Is it possible to construct in polynomial time the minimum dilation triangulation of a point set, or of a simple polygon Clearly there still also remains a wide gap between the best upper and lower bounds (2 and # 2 ....

M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. S. Tan. Edge-insertion for optimal triangulations. Disc. Comp. Geom., vol. 10, 1993, pp. 47--65.

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M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. S. Tan. Edge insertion for optimal triangulations. Discrete Comput. Geom., 10(1):47--65, 1993.

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M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. S. Tan, "Edge insertion for optimal triangulations", Discrete and Computational Geometry 10 (1993) 47-65.

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M. Bern, H. Edelsbrunner, D. Eppstein, S. Mitchell, and T. S. Tan. Edge-insertion for optimal triangulations. Discrete Comput. Geom., 10:47--65, 1993.

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