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122
The Quickhull algorithm for convex hulls
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1996
"... The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algo ..."
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Cited by 452 (0 self)
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The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floatingpoint arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.
Voronoi Diagrams and Delaunay Triangulations
 Computing in Euclidean Geometry
, 1992
"... The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi ..."
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Cited by 196 (3 self)
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The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi diagrams. 1 Introduction Let S be a set of n points in ddimensional euclidean space E d . The points of S are called sites. The Voronoi diagram of S splits E d into regions with one region for each site, so that the points in the region for site s2S are closer to s than to any other site in S. The Delaunay triangulation of S is the unique triangulation of S so that there are no elements of S inside the circumsphere of any triangle. Here `triangulation' is extended from the planar usage to arbitrary dimension: a triangulation decomposes the convex hull of S into simplices using elements of S as vertices. The existence and uniqueness of the Delaunay triangulation are perhaps not obvio...
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.
Tetrahedral Mesh Improvement Using Swapping and Smoothing
 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
, 1997
"... Automatic mesh generation and adaptive refinement methods for complex threedimensional domains have proven to be very successful tools for the efficient solution of complex applications problems. These methods can, however, produce poorly shaped elements that cause the numerical solution to be less ..."
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Cited by 86 (10 self)
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Automatic mesh generation and adaptive refinement methods for complex threedimensional domains have proven to be very successful tools for the efficient solution of complex applications problems. These methods can, however, produce poorly shaped elements that cause the numerical solution to be less accurate and more difficult to compute. Fortunately, the shape of the elements can be improved through several mechanisms, including face and edgeswapping techniques, which change local connectivity, and optimizationbased mesh smoothing methods, which adjust mesh point location. We consider several criteria for each of these two methods and compare the quality of several meshes obtained by using different combinations of swapping and smoothing. Computational experiments show that swapping is critical to the improvement of general mesh quality and that optimizationbased smoothing is highly effective in eliminating very small and very large angles. Highquality meshes are obtained in a computationally efficient manner by using optimizationbased smoothing to improve only the worst elements and a smart variant of Laplacian smoothing on the remaining elements. Based on our experiments, we offer several recommendations for the improvement of tetrahedral meshes.
An Approach to Combined Laplacian and OptimizationBased Smoothing for Triangular, Quadrilateral, and QuadDominant Meshes
 INTERNATIONAL MESHING ROUNDTABLE
, 1998
"... Automatic finite element mesh generation techniques have become commonly used tools for the analysis of complex, realworld models. All of these methods can, however, create distorted and even unusable elements. Fortunately, several techniques exist which can take an existing mesh and improve its qu ..."
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Cited by 54 (3 self)
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Automatic finite element mesh generation techniques have become commonly used tools for the analysis of complex, realworld models. All of these methods can, however, create distorted and even unusable elements. Fortunately, several techniques exist which can take an existing mesh and improve its quality. Smoothing (also referred to as mesh relaxation) is one such method, which repositions nodal locations, so as to minimize element distortion. In this paper, an overall mesh smoothing scheme is presented for meshes consisting of triangular, quadrilateral, or mixed triangular and quadrilateral elements. This paper describes an efficient and robust combination of constrained Laplacian smoothing together with an optimizationbased smoothing algorithm. The smoothing algorithms have been implemented in ANSYS and performance times are presented along with several example models.
Mesh Generation
 Handbook of Computational Geometry. Elsevier Science
, 2000
"... this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary. ..."
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Cited by 48 (6 self)
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this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary.
Nice Point Sets Can Have Nasty Delaunay Triangulations
 In Proc. 17th Annu. ACM Sympos. Comput. Geom
, 2001
"... We consider the complexity of Delaunay triangulations of sets of points in IR 3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of u points in ..."
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Cited by 48 (5 self)
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We consider the complexity of Delaunay triangulations of sets of points in IR 3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of u points in IR 3 with spread A has complexity il(min{A 3 , uA, u2}) and O (min{A 4, u2}). For the case A = D(v/), our lower bound construction consists of a gridlike sample of a right circular cylinder with constant height and radius. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has nearquadratic complexity.
On deletion in Delaunay triangulation
 Internat. J. Comput. Geom. Appl
, 2002
"... This paper presents how the space of spheres and shelling may be used to delete a point from a ddimensional triangulation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O(k log k), but we notice that this number only applies to low cost operations, while ..."
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Cited by 46 (4 self)
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This paper presents how the space of spheres and shelling may be used to delete a point from a ddimensional triangulation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O(k log k), but we notice that this number only applies to low cost operations, while time consuming computations are only done a linear number of times. This algorithm may be viewed as a variation of Heller’s algorithm,[1, 2] which is popular in the geographic information system community. Unfortunately, Heller algorithm is false, as explained in this paper.
Local OptimizationBased Simplicial Mesh Untangling And Improvement
 International Journal of Numerical Methods in Engineering
"... . We present an optimizationbased approach for mesh untangling that maximizes the minimum area or volume of simplicial elements in a local submesh. These functions are linear with respect to the free vertex position; thus the problem can be formulated as a linear program that is solved by using the ..."
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Cited by 46 (6 self)
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. We present an optimizationbased approach for mesh untangling that maximizes the minimum area or volume of simplicial elements in a local submesh. These functions are linear with respect to the free vertex position; thus the problem can be formulated as a linear program that is solved by using the computationally inexpensive simplex method. We prove that the function level sets are convex regardless of the position of the free vertex, and hence the local subproblem is guaranteed to converge. Maximizing the minimum area or volume of mesh elements, although wellsuited for mesh untangling, is not ideal for mesh improvement, and its use often results in poor quality meshes. We therefore combine the mesh untangling technique with optimizationbased mesh improvement techniques and expand previous results to show that a commonly used twodimensional mesh quality criterion can be guaranteed to converge when starting with a valid mesh. Typical results showing the effectiveness of the combine...