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42
Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 180 (8 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. ..."
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Cited by 143 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...
PiecewiseLinear Interpolation between Polygonal Slices
 Computer Vision and Image Understanding
, 1994
"... In this paper we present a new technique for piecewiselinear surface reconstruction from a series of parallel polygonal crosssections. This is an important problem in medical imaging, surface reconstruction from topographic data, and other applications. We reduce the problem, as in most previous wo ..."
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Cited by 65 (12 self)
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In this paper we present a new technique for piecewiselinear surface reconstruction from a series of parallel polygonal crosssections. This is an important problem in medical imaging, surface reconstruction from topographic data, and other applications. We reduce the problem, as in most previous works, to a series of problems of piecewiselinear interpolation between each pair of successive slices. Our algorithm uses a partial curve matching technique for matching parts of the contours, an optimal triangulation of 3D polygons for resolving the unmatched parts, and a minimum spanning tree heuristic for interpolating between non simply connected regions. Unlike previous attempts at solving this problem, our algorithm seems to handle successfully any kind of data. It allows multiple contours in each slice, with any hierarchy of contour nesting, and avoids the introduction of counterintuitive bridges between contours, proposed in some earlier papers to handle interpolation between multi...
Filling Gaps in the Boundary of a Polyhedron
 Computer Aided Geometric Design
, 1993
"... In this paper we present an algorithm for detecting and repairing defects in the boundary of a polyhedron. These defects, usually caused by problems in CAD software, consist of small gaps bounded by edges that are incident to only one polyhedron face. The algorithm uses a partial curve matching t ..."
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Cited by 38 (4 self)
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In this paper we present an algorithm for detecting and repairing defects in the boundary of a polyhedron. These defects, usually caused by problems in CAD software, consist of small gaps bounded by edges that are incident to only one polyhedron face. The algorithm uses a partial curve matching technique for matching parts of the defects, and an optimal triangulation of 3D polygons for resolving the unmatched parts. It is also shown that finding a consistent set of partial curve matches with maximum score, a subproblem which is related to our repairing process, is NPHard. Experimental results on several polyhedra are presented. Keywords: CAD, polyhedra, gap filling, curve matching, geometric hashing, triangulation. 1 Introduction The problem studied in this paper is the detection and repair of "gaps" in the boundary of a polyhedron. This problem usually appears in polyhedral approximations of CAD objects, whose boundaries are described using curved entities of higher leve...
Edge Insertion for Optimal Triangulations
, 1993
"... The edgeinsertion paradigm improves a triangulation of a finite point set in R² iteratively by adding a new edge, deleting intersecting old edges, and retriangulating the resulting two polygonal regions. After presenting an abstract view of the paradigm, this paper shows that it can be used to o ..."
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Cited by 28 (3 self)
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The edgeinsertion paradigm improves a triangulation of a finite point set in R² iteratively by adding a new edge, deleting intersecting old edges, and retriangulating the resulting two polygonal regions. After presenting an abstract view of the paradigm, this paper shows that it can be used to obtain polynomial time algorithms for several types of optimal triangulations.
A Quadratic Time Algorithm for the MinMax Length Triangulation
 SIAM J. Comput
, 1991
"... Abstract. We show that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time O(n 2). The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative n ..."
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Cited by 28 (3 self)
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Abstract. We show that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time O(n 2). The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative neighborhood graph of the points as a subgraph. With minor modi cations the algorithm works for arbitrary normed metrics. Key words. Computational geometry,point sets, triangulations, two dimensions, minmax edge length, normed metrics AMS(MOS) subject classi cations. 68U05, 68Q25, 65D05 Appear in: SIAM Journal on Computing, 22 (3), 527{551, (1993)
Arc triangulations
 PROC. 26TH EUR. WORKSH. COMP. GEOMETRY (EUROCG’10)
, 2010
"... The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alter ..."
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Cited by 27 (2 self)
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The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alternative that offers flexibility for additionally enlarging small angles. We show that angle optimization and related questions lead to linear programming problems, and we define unique flips in arc triangulations. Moreover, applications of certain classes of arc triangulations in the areas of finite element methods and graph drawing are sketched.
AN O(n² log n) TIME ALGORITHM FOR THE Minmax Angle Triangulation
"... We show that a triangulation ofasetofn points in the plane that minimizes the maximum angle can be computed in time O(n² log n) and space O(n). The algorithm is fairly easy to implement and is based on the edgeinsertionscheme that iteratively improves an arbitrary initial triangulation. It can be ..."
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Cited by 23 (4 self)
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We show that a triangulation ofasetofn points in the plane that minimizes the maximum angle can be computed in time O(n² log n) and space O(n). The algorithm is fairly easy to implement and is based on the edgeinsertionscheme that iteratively improves an arbitrary initial triangulation. It can be extended to the case where edges are prescribed, and, within the same time and spacebounds, it can lexicographically minimize the sorted angle vector if the point setis in general position. Experimental results on the e ciency of the algorithm and the quality ofthe triangulations obtained are included.
Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
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Cited by 19 (3 self)
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We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
Aggressive Tetrahedral Mesh Improvement
 In Proc. of the 16th Int. Meshing Roundtable
, 2007
"... Summary. We present a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or “mesh cleanup. ” Our goal is to aggressively optimize the worst t ..."
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Cited by 17 (4 self)
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Summary. We present a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or “mesh cleanup. ” Our goal is to aggressively optimize the worst tetrahedra, with speed a secondary consideration. Mesh optimization methods often get stuck in bad local optima (poorquality meshes) because their repertoire of mesh transformations is weak. We employ a broader palette of operations than any previous mesh improvement software. Alongside the best traditional topological and smoothing operations, we introduce a topological transformation that inserts a new vertex (sometimes deleting others at the same time). We describe a schedule for applying and composing these operations that rarely gets stuck in a bad optimum. We demonstrate that all three techniques—smoothing, vertex insertion, and traditional transformations—are substantially more effective than any two alone. Our implementation usually improves meshes so that all dihedral angles are between 31 ◦ and 149 ◦ , or (with a different objective function) between 23 ◦ and 136 ◦. 1