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31
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 179 (7 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.
Competitive Online Routing in Geometric Graphs
 Theoretical Computer Science
, 2001
"... We consider online routing algorithms for finding paths between the vertices of plane graphs. ..."
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Cited by 34 (4 self)
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We consider online routing algorithms for finding paths between the vertices of plane graphs.
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.
Simple Traversal of a Subdivision Without Extra Storage
 International Journal of Geographic Information Systems
, 1996
"... In this paper we show how to traverse a subdivision and to report all cells, edges and vertices, without making use of mark bits in the structure or a stack. We do this by performing a depthfirst search on the subdivision, using local criteria for deciding what is the next cell to visit. Our method ..."
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Cited by 27 (2 self)
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In this paper we show how to traverse a subdivision and to report all cells, edges and vertices, without making use of mark bits in the structure or a stack. We do this by performing a depthfirst search on the subdivision, using local criteria for deciding what is the next cell to visit. Our method is extremely simple and provably correct. The algorithm has applications in the field of Geographic Information Systems (GIS), where traversing subdivisions is a common operation, but modifying the database is unwanted or impossible. We show how to adapt our algorithm to answer related queries, such as windowing queries and reporting connected subsets of cells that have a common attribute. Finally, we show how to extend our algorithm such that it can handle convex 3dimensional subdivisions. Keywords: Subdivisions, traversal algorithms, topological data structure, windowing, three dimensions. 1 Introduction The basic spatial vector data structure in any geographic information system is the...
Triangulating Polygons Without Large Angles
, 1995
"... We show how to triangulate polygonal regionsadding extra vertices as necessary with triangles of guaranteed quality. Using only O(n) triangles, we can guarantee that the smallest height (shortest dimension) of a triangle in a triangulation of an nvertex polygon (with holes) is a constant f ..."
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Cited by 26 (4 self)
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We show how to triangulate polygonal regionsadding extra vertices as necessary with triangles of guaranteed quality. Using only O(n) triangles, we can guarantee that the smallest height (shortest dimension) of a triangle in a triangulation of an nvertex polygon (with holes) is a constant fraction of the largest possible. For simple polygons, using O(n log n) triangles, we can guarantee that the largest angle is no greater than 150 ffi . This bound increases to O(n 3=2 ) triangles for the case of polygons with holes. We can add the guarantee on smallest height to these nolargeangle results, without increasing the asymptotic complexity of the triangulation. Finally we give a nonobtuse triangulation algorithm for convex polygons that uses O(n 1:85 ) triangles. Keywords: Computational geometry, mesh generation, triangulation, angle condition. 1. Introduction There have been a number of recent papers on the general problem of triangulating a planar point set or pol...
A OneStep Crust and Skeleton Extraction Algorithm
 Algorithmica
, 2003
"... We wish to extract the topology from scanned maps. In previous work [GNY96] this was done by extracting a skeleton from the Voronoi diagram, but this required vertex labelling and was only useable for polygon maps. We wished to take the crust algorithm of Amenta, Bern and Eppstein [ABE98] and modify ..."
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Cited by 17 (9 self)
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We wish to extract the topology from scanned maps. In previous work [GNY96] this was done by extracting a skeleton from the Voronoi diagram, but this required vertex labelling and was only useable for polygon maps. We wished to take the crust algorithm of Amenta, Bern and Eppstein [ABE98] and modify it to extract the skeleton from unlabelled vertices. We find that by reducing the algorithm to a local test on the original Voronoi diagram we may extract both a crust and a skeleton simultaneously, using a variant of the QuadEdge structure of [GS85]. We show that this crust has the properties of the original, and that the resulting skeleton has many practical uses. We illustrate the usefulness of the combined diagram with various applications.
An Improved Algorithm for Subdivision Traversal without Extra Storage
, 2000
"... We describe an algorithm for enumerating all vertices, edges and faces of a planar subdivision stored in any of the usual pointerbased representations, while using only a constant amount of memory beyond that required to store the subdivision. The algorithm is a refinement of a method introduced ..."
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Cited by 13 (2 self)
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We describe an algorithm for enumerating all vertices, edges and faces of a planar subdivision stored in any of the usual pointerbased representations, while using only a constant amount of memory beyond that required to store the subdivision. The algorithm is a refinement of a method introduced by de Berg et al (1997), that reduces the worst case running time from O(n²) to O(n log n). We also give experimental results that show that our modified algorithm runs faster not only in the worst case, but also in many realistic cases.
CONSTANTWORKSPACE ALGORITHMS FOR GEOMETRIC PROBLEMS
 JOURNAL OF COMPUTATIONAL GEOMETRY
, 2011
"... Constantworkspace algorithms may use only constantly many cells of storage in addition to their input, which is provided as a readonly array. We show how to construct several geometric structures efficiently in the constantworkspace model. Traditional algorithms process the input into a suitabl ..."
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Cited by 7 (3 self)
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Constantworkspace algorithms may use only constantly many cells of storage in addition to their input, which is provided as a readonly array. We show how to construct several geometric structures efficiently in the constantworkspace model. Traditional algorithms process the input into a suitable data structure (like a doublyconnected edge list) that allows efficient traversal of the structure at hand. In the constantworkspace setting, however, we cannot afford to do this. Instead, we provide operations that compute the desired features on the fly by accessing the input with no extra space. The whole geometric structure can be obtained by using these operations to enumerate all the features. Of course, we must pay for the space savings by slower running times. While the standard data structure allows us to implement traversal operations in constant time, our schemes typically take linear time to read the input data in each step. We begin with two simple problems: triangulating a planar point set and finding the trapezoidal decomposition of a simple polygon. In both cases adjacent features can be enumerated in linear time per step, resulting in total quadratic running time to output the whole structure. Actually, we show that the former result carries over to the Delaunay triangulation, and hence the Voronoi diagram. This also means that we can compute the largest empty circle of a planar point set in quadratic time and constant workspace. As another application, we demonstrate how to enumerate the features of an Euclidean minimum spanning tree (EMST) in quadratic time per step, so that the whole EMST can be found in cubic time using constant workspace. Finally, we describe how to compute a shortest geodesic path between two points in a simple polygon. Although the shortest path problem in general graphs is NLcomplete [18], this constrained problem can be solved in quadratic time using only constant workspace.
Route Discovery with Constant Memory in Oriented Planar Geometric Networks
 In Algosensors
, 2004
"... We address the problem of discovering routes in strongly connected planar geometric networks with directed links. We consider two types of directed planar geometric networks: Eulerian (in which every vertex has the same number of ingoing and outgoing edges) and outerplanar (in which a single fac ..."
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Cited by 6 (4 self)
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We address the problem of discovering routes in strongly connected planar geometric networks with directed links. We consider two types of directed planar geometric networks: Eulerian (in which every vertex has the same number of ingoing and outgoing edges) and outerplanar (in which a single face contains all the vertices of the network). Motivated by the necessity for establishing communication in wireless networking based only on geographic proximity, in both instances we give algorithms that use only Escuela de Ciencias FisicoMatemticas de la Universidad Michoacana de San Nicols de Hidalgo, Mxico.