J. Erickson. Nice point sets can have nasty Delaunay triangulations. Proc. 17th ACM Sympos. Comput. Geom. (SoCG 01), 96--105, 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the sampling has to be uniform everywhere. We chose the lower bound on as 2 for making further calculations precise though any other constant greater than 1 will be equally valid for our theoretical analysis. Such sampling conditions have been studied in the context of surface reconstruction in [9, 15]. 2.2 Tangent and normal spaces. Since we are dealing with a collection of smooth manifolds in , results from differential geometry ensure that a tangent space at each M is well defined [7] The dimension of the tangent space at p coincides with the topological dimension of the manifold ....

J. Erickson. Nice point sets can have nasty Delaunay triangulations. Proc. 17th Ann. Sympos. Comput. Geom., (2001), 96--105.

....= n. The same holds where is the set of halfspaces in R , for d 3. Proof. Take P to be a set of n points on the positive branch of the moment curve # = t, t , t ) t . It is easy to verify that any pair of points p, q P are connected in the Delaunay triangulation of P [9], implying that there exists a ball hose intersection with P is p, q . Thus, all points must be colored using di#erent colors. The second claim follows by lifting P into the standard paraboloid in R by the map (x, y, z) ## (x, y, z, x y z ) A ball in R is mapped to ....

J. Erickson. Nice point sets can have nasty delaunay triangulations. In Proc. 17th Annu. ACM Sympos. Comput. Geom., pages 96--105, 2001.

....nor a subquadratic worst case running time (one may argue that their running time is linear under very restrictive uniformity conditions) One can ask whether the worst case quadratic size of Delaunay triangulations can occur for a set of samples from a smooth surface. Recent work of Erickson [16] shows that there are smooth surfaces with uniform sets of samples that have a Delaunay tetrahedrization of quadratic complexity. Therefore, even if the original COCONE algorithm uses an output sensitive algorithm for computing the Delaunay tetrahedrization [9] it could not match the running time ....

J. Erickson. Nice point sets can have nasty Delaunay triangulations. Available from the author's web site at www.cs.uiuc.edu. Proc. 17th ACM Sympos. Comput. Geom., 96--105, 2001.

....nor a subquadratic worstcase running time (one may argue that their running time is linear under very restrictive uniformity conditions) One can ask whether the worst case quadratic size of Delaunay triangulations can occur 1 for a set of samples from a smooth surface. Recent work of Erickson [16] shows that there are smooth surfaces with uniform sets of samples that have a Delaunay tetrahedrization of quadratic complexity. Therefore, even if the original COCONE algorithm uses an output sensitive algorithm for computing the Delaunay tetrahedrization [9] it could not match the running time ....

....construction of the Delaunay tetrahedrization has a quadratic worst case running time, so does the COCONE algorithm. A more careful analysis is required since it could be the case that the quadratic behavior do not occur for a set sampling a smooth surface. This has been investigated by Erickson [16]. For example, he describes how the usual quadratic construction consisting of points on two line segments can be embedded on a smooth surface; see Fig. 5. This example also shows that the number of candidate triangles can be quadratic in the worst case, so it is not feasible to obtain a better ....

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J. Erickson. Nice point sets can have nasty Delaunay triangulations. Available from the author's web site at www.cs.uiuc.edu. Proc. 17th ACM Sympos. Comput. Geom., 96--105, 2001.

....In Section 3, we describe a new implementation of the Cocone algorithm that runs in time O(n log n) under a locally uniform sampling condition, in addition to the original sampling condition. Such a condition seems reasonable for the output of current scanning techniques. Recent work of Erickson [8] shows that there are smooth surfaces with uniform sets of samples that have a Delaunay tetrahedrization of quadratic complexity. Therefore, even if the original Cocone algorithm uses an output sensitive algorithm for computing the Delaunay tetrahedrization [5] it could not match the running time ....

....at c and radius 2r contains at most 4 m samples. Intuitively, this conditions describes that the sampling density must not change too rapidly within a short distance on the surface. It does not imply an absolute upper bound on the sampling density anywhere, nor a uniform sampling as defined in [8] (but a sample set that is uniform in [8] is also locally uniform) Observation 1 Let c 2 S and r lfs(c) If the ball B centered at c with radius r is empty of samples, then the sphere B 0 centered at c with radius b r, b 2, contains at most m 0 (4 ) 2dlog 2 be samples. 3.2 ....

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J. Erickson. Nice point sets can have nasty Delaunay triangulations. Available from the author's web site at www.cs.uiuc.edu. Submitted to the 17th Annual ACM Symposium on Computational Geometry 2001

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J. Erickson. Nice point sets can have nasty Delaunay triangulations. Proc. 17th Annu. ACM Sympos. Comput. Geom., 96--105, 2001. # je#e/pubs/spread.html#.

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J. Erickson. Nice point sets can have nasty Delaunay triangulations. Discrete Comput. Geom. 30(1):109--132, 2003.

....and Ramos [18, 23] is a notable exception. Since three dimensional Delaunay triangulations can have complexity # n in the worst case, these algorithms have worst case running time# ) However, this behavior is almost never observed in practice [12, 17] except for highly contrived inputs [21]. For all practical purposes, Delaunay triangulations of surface points appear to have linear complexity. The first subquadratic complexity bound for Delaunay triangulations of surface points was obtained by Golin and Na [25, 26, 27] They proved that if n points are chosen uniformly at random on ....

....on geometric parameters of the fixed surface, such as the number of facets, angles between adjacent edges, and angles between facet planes. For this reason, none of these bounds apply to smooth surfaces. Previously known bounds for non polyhedral surfaces are much weaker. In two earlier papers [21, 22], we analyzed the complexity of three dimensional Delaunay triangulations in terms of a geometric parameter called the spread, defined as the ratio between the largest and smallest pairwise distances. Our results imply that any (#, k) sample of any fixed (not necessarily polyhedral or smooth) ....

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J. Erickson. Nice point sets can have nasty Delaunay triangulations. Proc. 17th Annu. ACM Sympos. Comput. Geom., 96--105, 2001. # je#e/pubs/spread.html#.

....meshes, local selfintersecting polyhedra, or more generally, any local simplex soup whose edge graph is connected. For disconnected sets of simplices, however, our running time analysis requires that the spread of the vertices the ratio between the largest and smallest pairwise distance [13] is bounded by a polynomial in n. 4. THE HARPSICORDION We now show that the results from the previous section are asymptotically optimal for polyhedra in IR by constructing a pair of local polyhedra that intersect in n log n) distinct points. Our lower bound construction also implies that ....

J. Erickson. Nice point sets can have nasty Delaunay triangulations. Proc. 17th Annu. ACM Sympos. Comput. Geom., 96--105, 2001.

....Large Neighborly Families of Congruent Symmetric Convex 3 Polytopes Je Erickson University of Illinois at Urbana Champaign je e cs.uiuc.edu http: www.cs. uiuc.edu je e June 12, 2001 Abstract We construct, for any positive integer n, a family of n congruent convex polyhedra in IR 3 , such that every pair intersects in a common facet. Previously, the largest such family contained only eight polytopes. Our polyhedra are Voronoi regions of evenly distributed points on the ....

....convex polytopes in IR d , any dd=2e of which share a unique common boundary face. We also introduce a new family of cyclic polytopes, generalizing both the classic cyclic polytopes and the Petrie polytopes. 2 The Main Theorem Our construction relies on the following observation of the author [10] We include the proof for the sake of completeness. Lemma 1. Let (t) denote the unique sphere passing through h(t) and h( t) and tangent to the helix at those two points. For any 0 t pi, the sphere (t) intersects the helix only at its two points of tangency. Proof: Since a 180 degree ....

J. Erickson. Nice point sets can have nasty Delaunay triangulations. Proc. 17th Annu. ACM Sympos. Comput. Geom., pp. 96-105, 2001. arXiv:cs.CG/0103017.

....appear to have linear complexity. This frustrating discrepancy between theory and practice motivates our investigation of practical geometric constraints that imply low complexity Delaunay triangulations. Previous works in this direction have studied random point sets under various distributions [28, 27, 35, 38]; well spaced point sets, which are lowdiscrepancy samples of Lipschitz density functions [20, 49, 51, 52] and surface samples with various density Portions of this work were done while the author was visiting The Ohio State University. This research was partially supported by a Sloan ....

.... Portions of this work were done while the author was visiting The Ohio State University. This research was partially supported by a Sloan Fellowship and by NSF CAREER grant CCR 0093348. See http: www.cs.uiuc.edu je e pubs screw.html for the most recent version of this paper. constraints [8, 35]. We will discuss the connections between these models and our results shortly. Our e orts fall under the rubric of realistic input models, which have been primarily studied for inputs consisting of polygons or polyhedra [10, 63] This paper investigates the complexity of threedimensional ....

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J. Erickson. Nice point sets can have nasty Delaunay triangulations. Proc. 17th Annu. ACM Sympos. Comput. Geom., 96-105, 2001. hhttp://www.cs.uiuc.edu/~jee/pubs/spread.htmli.

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J. Erickson. Nice point sets can have nasty Delaunay triangulations. Proc. 17th ACM Sympos. Comput. Geom. (SoCG 01), 96--105, 2001.

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Je Erickson. Nice point sets can have nasty Delaunay triangulations. In Proc. 17th Annu. ACM Sympos. Comput. Geom., pages 96-105, 2001.

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J. Erickson. Nice point sets can have nasty Delaunay Triangulations. Proc. 17th Annu. Sympos. Comput. Geom., pp 96-105, 2001.

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J. Erickson. Nice points sets can have nasty delaunay triangulations. In Proc. 17th Annu. ACM Sympos. Comput. Geom., pages 96-105, 2001.

No context found.

J. Erickson. Nice point sets can have nasty delaunay triangulations. Discrete Comput. Geom., 30(1):109--132, 2003.

No context found.

Je Erickson. Nice point sets can have nasty Delaunay triangulations. In Proc. 17th Annu. ACM Sympos. Comput. Geom., pages 96105, 2001.

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J. Erickson. Nice point sets can have nasty Delaunay triangulations. Proc. 17th ACM Sympos. Comput. Geom. (SoCG 01), 96--105, 2001.

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Jeff Erikson, "Nice Point Sets Can have Nasty Delaunay Triangulations," Proceedings of the Seventeenth Annual ACM Symposium on Computational Geometry, (June 3-5, 2001) 96-105.

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