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Using triangles to partition a disk

1990
*
Discrete Mathematics
*

Any

doi:10.1016/0012-365x(90)90378-u
fatcat:q3i6cka4dfd65nh5sagiwt7nla
*partition*of*a**disk*into n subdisks can be imbedded in E* by laying down n*triangles*one after another. This imbedding can also be constructed by drawing 2n + 1 line segments. ... Introduction We will show how*to*imbed*a*given*partition*of*a**disk*into n subdisks in E2 by laying down n*triangles*, one after another, as in Fig. 1 or 2. ... Every*partition*of*a**disk*into n subdisks can be imbedded in E2 by drawing 2n + 1 line segments. Proof.*Use*T, and B,, . . . , B,. 0 2 .Fig. 1 Lemma 2. 2 . 212 Shingled imbeddings Theorem 2.1. ...##
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Minimal Partitions for p-norms of Eigenvalues
[article]

2018
*
arXiv
*
pre-print

In this article we are interested in studying

arXiv:1612.07296v2
fatcat:lvcq4g4rfnczphiddfx4gywioi
*partitions*of the square, the*disk*and the equilateral*triangle*which minimize*a*p-norm of eigenvalues of the Dirichlet-Laplace operator. ... This allows*us**to*see when*partitions*minimizing the 1-norm and the infinity-norm are different. ... The authors thank Michael Floater for suggesting*us**to*look more carefully at k-*partitions*of the equilateral*triangle*when k is*a*triangular number. ...##
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Minimal partitions for $p$-norms of eigenvalues

2018
*
Interfaces and free boundaries (Print)
*

In this article we are interested in studying

doi:10.4171/ifb/399
fatcat:danq3lvolrezrj6v57mdoe3ote
*partitions*of the square, the*disk*and the equilateral*triangle*which minimize*a*p-norm of eigenvalues of the Dirichlet-Laplace operator. ... This allows*us**to*see when*partitions*minimizing the 1-norm and the infinity-norm are different. ... The authors thank Michael Floater for suggesting*us**to*look more carefully at k-*partitions*of the equilateral*triangle*when k is*a*triangular number. ...##
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Optimal partitions for the sum and the maximum of eigenvalues
[article]

2017
*
arXiv
*
pre-print

We analyze in detail the square, the

arXiv:1702.01567v1
fatcat:zpcl4sl3nvdijl5kygiiffecfi
*disk*and the equilateral*triangle*. ... In this paper we compare the candidates*to*be spectral minimal*partitions*for two criteria: the maximum and the average of the first eigenvalue on each subdomains of the*partition*. ... Acknowledgements This work is supported by*a*public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (LabEx Sciences Mathématiques de Paris ...##
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Triangle listing in massive networks and its applications

2011
*
Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '11
*

We propose an I/O-efficient algorithm for

doi:10.1145/2020408.2020513
dblp:conf/kdd/ChuC11
fatcat:jmidmuy465av5auzdes6pc2jri
*triangle*listing. Our algorithm is exact and avoids random*disk*access. ... When the input graph cannot fit into main memory,*triangle*listing requires random*disk*accesses that can incur prohibitively large I/O cost. ... Algorithm 1 In-Memory*Triangle*Listing Input:*A*graph G = (VG, EG) Output: △(G) Acknowledgment The authors would like*to*thank the reviewers for their constructive comments. ...##
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Optimal Data-Space Partitioning of Spatial Data for Parallel I/O

2005
*
Distributed and parallel databases
*

Some of these results carry over

doi:10.1023/b:dapd.0000045550.56749.75
fatcat:yfsq7rhkibegpdytmilvmoxoyq
*to*circular*partitioning*of the data-which is an example of*a*non-convex region. ... This paper explores optimal*partitioning*for two-dimensional data for*a*class of queries and develops multi-*disk*allocation techniques that maximize the degree of I/O parallelism obtained in each case. ... Given the number M of available*disks*, we*use**a*generic allocation technique parameterized by*a*skip value H . The*partition*(0, 0) are assigned*to*device 0. ...##
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The Use of Precomputed Triangle Clusters for Accelerated Ray Tracing in Dynamic Scenes

2009
*
Computer graphics forum (Print)
*

This algorithm precomputes densely packed clusters of

doi:10.1111/j.1467-8659.2009.01497.x
fatcat:j77dyw6wa5g5no3vqpccmxwnuy
*triangles*on surfaces. Folowing that,*a*set of clusters is*used**to*rebuild the BVH in every frame. ... We introduce*a*novel heuristic*to*create*triangle*clusters that are designed for high performance ray tracing. ... Any cluster should have*a*shape similar*to**a*sphere or*to**a**disk*. ...##
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A novel page-based data structure for interactive walkthroughs

2009
*
Proceedings of the 2009 symposium on Interactive 3D graphics and games - I3D '09
*

Assume that each

doi:10.1145/1507149.1507153
dblp:conf/si3d/SajadiHDYG09
fatcat:jnfgdxqcubcihntj236ecqbwtq
*disk*-page consists of*a*set of*triangles*whose bounding boxes are computed. ... Walkthrough experiments have been conducted on*a*city scene consisting of over 110M*triangles*. ...*A**disk*page K-d tree, for example, is*a*K-d tree on these bounding boxes of the*disk*pages. The leaf nodes of the K-d tree will have pointers*to*the*disk*-pages, and not individual*triangles*. ...##
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Bi-Lipschitz equivalent Alexandrov surfaces, II
[article]

2004
*
arXiv
*
pre-print

This is

arXiv:math/0409343v1
fatcat:n67k3lwbp5grxcprxlpr2vcxa4
*a*continuation of the joint paper with the same title by A.Belenkiy and Yu.Burago. ... integral curvatures, and two positive numbers e and l such that positive curvature of each embedded*disk*with perimeter not greater than l is not greater than \pi-e. ... This allows*us**to*replace each*triangle*△BA i*A*i+1 by*a**triangle*flat outside the*disk*D (B, r) and L-bi-Lipschitz equivalent*to*△BA i*A*i+1 . ...##
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Organizing and indexing non-convex regions

2008
*
Proceedings of the VLDB Endowment
*

The index defines

doi:10.14778/1454159.1454212
fatcat:gexwexqgirf2dkkk5poc3mhw4i
*a*linear order on the Delaunay*triangles*. Data from each*triangle*are then placed on*disk*in index order, e.g.*using**a*B+-tree. ... We*use*computational geometry techniques*to*automatically characterize the region of space from which data are drawn,*partition*the region based on that characterization, and create an index from the*partitions*... The index defines*a*linear order on the Delaunay*triangles*. Data from each*triangle*are then placed on*disk*in index order, e.g.*using**a*B+-tree. ...##
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Computational prediction of electrical and thermal conductivities of disklike particulate composites

2015
*
International Journal of Computational Materials Science and Engineering
*

The pairwise intersection criteria of

doi:10.1142/s204768411550013x
fatcat:4unebapj5bhfloyluzdvudpqz4
*disks*are developed*using**a*set of vector operations. ... An element*partition*scheme has been implemented*to*connect the elements on different*disks*across the lines of intersection. ... Acknowledgment The authors would like*to*acknowledge support from the National Science Foundation under contract number #1232520. ...##
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Out-Of-Core Sort-First Parallel Rendering for Cluster-Based Tiled Displays
[article]

2002
*
Eurographics Symposium on Parallel Graphics and Visualization
*

Given

doi:10.2312/egpgv/egpgv02/089-096
fatcat:67cakk4qazdu5klsqvitio5tqe
*a*model, we*use*an out-of-core preprocessing algorithm*to*build an on-*disk*hierarchical representation for the model. ... At run time, each PC renders the image for*a*display tile,*using*an out-of-core rendering approach that employs multiple threads*to*overlap rendering, visibility computation, and*disk*operations. ...*To*achieve*a*well balanced system, they developed dynamic screen*partitioning*schemes that predict the rendering costs of groups of*triangles*and attempt*to*minimize the amount of overlap between*triangles*...##
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Scalable isosurface visualization of massive datasets on commodity off-the-shelf clusters

2009
*
Journal of Parallel and Distributed Computing
*

It statically

doi:10.1016/j.jpdc.2008.07.006
pmid:19756231
pmcid:PMC2743442
fatcat:od7t7rkw7rak5btqvvrcm3is6i
*partitions*the volume data*to*parallel*disks*with*a*balanced workload spectrum, and builds I/O-optimal external interval trees*to*minimize the number of I/O operations of loading large data ... It achieves scalability by*using*both parallel and out-of-core processing and parallel*disks*. ... Acknowledgments This research is supported in part by grants ACI-9982297, DMS-9873326 from the National Science Foundation,*a*DOE-ASCI grant BD4485-MOID from Sandia National Laboratory, Lawrence Livermore ...##
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Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition
[chapter]

2013
*
Lecture Notes in Computer Science
*

Let D be

doi:10.1007/978-3-642-40104-6_42
fatcat:3dywiztwgze6jdnimmzc7hxgay
*a*set of n pairwise disjoint unit*disks*in the plane. 8 We describe how*to*build*a*data structure for D so that for any point 9 set P containing exactly one point from each*disk*, we can quickly ... We also provide*a*matching lower bound. 14 Our solution is based on*a*recursive space decomposition, combined with 15*a*fast algorithm*to*compute the union of two disjoint onion decomposi-16 tions. ... . , ∆*a*be the resulting*triangles*and Q = 39 Q 1∪ · · ·∪Q*a*the*partition*of Q. ...##
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Unions of Onions: Preprocessing Imprecise Points for Fast Onion Decomposition
[article]

2013
*
arXiv
*
pre-print

Let D be

arXiv:1302.5328v1
fatcat:f65o3x33bjb5lg2emrm6a4olku
*a*set of n pairwise disjoint unit*disks*in the plane. ... We describe how*to*build*a*data structure for D so that for any point set P containing exactly one point from each*disk*, we can quickly find the onion decomposition (convex layers) of P. ...*A*30 simplicial r-*partition*of X is*a*sequence ∆ 1 , . . . , ∆ r of r*triangles*and*a**partition*31 X = X 1∪ · · ·∪X r of X into r pieces such that (i) for i = 1, . . . , r, we have X i ⊆ 32 ∆ i and |X ...
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