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Reporting Intersecting Pairs of Polytopes in Two and Three Dimensions [chapter]

Pankaj K. Agarwal, Mark de Berg, Sariel Har-Peled, Mark H. Overmars, Micha Sharir, Jan Vahrenhold
2001 Lecture Notes in Computer Science  
Our algorithms can be modified to count the number of intersecting pairs in O(n 4/3 log O(1) n) time for the planar case, and in O(n 8/5+ε ) time and R 3 .  ...  We present output-sensitive algorithms for reporting all k pairs of indices (i, j) such that Pi intersects Pj.  ...  Conclusions In this paper, we presented output-sensitive algorithms for reporting all intersecting pairs of convex polygons / polytopes in two and three dimensions.  ... 
doi:10.1007/3-540-44634-6_12 fatcat:jd2ynu3h4zef7k463fiavapazi

Reporting intersecting pairs of convex polytopes in two and three dimensions

Pankaj K. Agarwal, Mark de Berg, Sariel Har-Peled, Mark H. Overmars, Micha Sharir, Jan Vahrenhold
2002 Computational geometry  
Our algorithms can be modified to count the number of intersecting pairs in O(n 4/3 log 2+ε n) time for the planar case, and in O(n 8/5+ε ) time for the three-dimensional case.   ...  We present output-sensitive algorithms for reporting all k pairs of indices (i, j ) such that P i intersects P j .  ...  Conclusions In this paper, we presented output-sensitive algorithms for reporting all intersecting pairs among a set of convex polygons in the plane, and among a set of convex polytopes in three dimensions  ... 
doi:10.1016/s0925-7721(02)00049-4 fatcat:wl2swhgeubdfrnh3oyg7gm6kr4

Splitting a complex of convex polytopes in any dimension

Chandrajit L. Bajaj, Valerio Pascucci
1996 Proceedings of the twelfth annual symposium on Computational geometry - SCG '96  
In Figure 3 two weak complexes of intrinsic dimension two (a) and three (b) are depicted.  ...  Definition 2 A set of convex polytopes forms a weak complex WCif, for each pair of polytopes c and g,the intersection f = c \ g satisfies one of the following two conditions: 1. c \ g = ; 2. c\g = @c\@  ...  Split the polytopes with a hyperplane h that forms the same angle with f 1 and f 2 and passes through the (k ;2)-polytopes that they have in common.  ... 
doi:10.1145/237218.237246 dblp:conf/compgeom/BajajP96 fatcat:zjo2zqee2nfv5nw4wmaeuveicu

Conservative interpolation between volume meshes by local Galerkin projection

P.E. Farrell, J.R. Maddison
2011 Computer Methods in Applied Mechanics and Engineering  
This paper presents an algorithm for the local implementation of Galerkin projection of discrete fields between meshes. This algorithm extends naturally to three dimensions and is very efficient.  ...  The problem of interpolating between discrete fields arises frequently in computational physics.  ...  Farrell would like to thank AWE for their funding of his research through the Institute of Shock Physics.  ... 
doi:10.1016/j.cma.2010.07.015 fatcat:wdudtrxfaff5tg547vvww623ve

I-COLLIDE

Jonathan D. Cohen, Ming C. Lin, Dinesh Manocha, Madhav Ponamgi
1995 Proceedings of the 1995 symposium on Interactive 3D graphics - SI3D '95  
The algorithm uses a two-level approach based on pruning multipleobject pairs using bounding boxes and performing exact collision detection between selected pairs of polyhedral models.  ...  In particular, the system takes less than l/20 of a second to determine all the collisions and contacts in an environment consisting of more than a 1000 moving polytopes, each consisting of more than 50  ...  ACKNOWLEDGEMENTS We are grateful to John Canny and David Baraff for productive discussions and to Brian Mirtich for his help in implementation of the convex polytope pair algorithm.  ... 
doi:10.1145/199404.199437 dblp:conf/si3d/CohenLMP95 fatcat:4ehyercrijfchiyjsjdsufjcvu

Translational packing of arbitrary polytopes

Jens Egeblad, Benny K. Nielsen, Marcus Brazil
2009 Computational geometry  
Additional details are given for the three-dimensional case and results are reported for the problem of packing polyhedra in a rectangular parallelepiped.  ...  For two polytopes with complexity O (n) and O (m) and a fixed dimension, the running time is O (nm log(nm)) for both the minimization and maximization variants of the translation algorithm.  ...  [13] and proved very successful for two dimensions. Three problem instances in three dimensions, by Stoyan et al.  ... 
doi:10.1016/j.comgeo.2008.06.003 fatcat:if6hpdq2o5aelaa4bswxxw5kyy

Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

David Eppstein, Maarten Löffler
2013 Discrete & Computational Geometry  
For any fixed d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time  ...  We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D.  ...  Acknowledgments This work was supported in part by NSF Grant 0830403 and by the Office of Naval Research under Grant N00014-08-1-1015.  ... 
doi:10.1007/s00454-013-9503-3 fatcat:hpbcxdboi5f5fmrhweup63cbbi

Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space

Naama Mayer, Efi Fogel, Dan Halperin
2010 Proceedings of the 14th ACM Symposium on Solid and Physical Modeling - SPM '10  
Abstract We present a novel algorithm for maintaining exact Minkowski sums of pairs of polytopes in R 3 , while one of the polytopes rotates.  ...  We give tight combinatorial bounds on the complexity of the criticality map when one of the polytopes rotates about one, two, or three axes.  ...  Chapter 3 defines the Minkowski-sum induced criticality maps in one, two, and three dimensions.  ... 
doi:10.1145/1839778.1839780 dblp:conf/sma/MayerFH10 fatcat:lozk3gn3ovfebpy3sbpzbpfpyq

Compact hyperbolic Coxeter five-dimensional polytopes with nine facets [article]

Jiming Ma, Fangting Zheng
2022 arXiv   pre-print
In this paper, we obtain a complete classification of compact hyperbolic Coxeter five-dimensional polytopes with nine facets.  ...  sets of three / four / five facets of which the intersection is not a face /an edge / a vertex of P k ,and no disjoint pairs is included. (7) The set e 3 / e 4 / e 5 of sets of three / four / five facets  ...  It was proved by Vinberg [Vin85 (1) ] that no compact hyperbolic Coxeter polytope exists in dimensions d ≥ 30; and no noncompact hyperbolic Coxeter polytope of finite volume exists in dimensions d ≥ 996  ... 
arXiv:2203.16049v1 fatcat:zfxwxoodeveabcycs5jh2tjdqq

Asymmetric Convex Intersection Testing

Luis Barba, Wolfgang Mulzer, Michael Wagner
2018 ACM-SIAM Symposium on Discrete Algorithms  
We consider asymmetric convex intersection testing (ACIT). Let P ⊂ R d be a set of n points and H a set of n halfspaces in d dimensions.  ...  We denote by ch(P ) the polytope obtained by taking the convex hull of P , and by fh(H) the polytope obtained by taking the intersection of the halfspaces in H.  ...  Convex polytopes in dimension d can be implicitly represented in two ways, either by its set of vertices, or by the set of halfspaces whose intersection defines the polytope.  ... 
doi:10.4230/oasics.sosa.2019.9 dblp:conf/soda/BarbaM19 fatcat:nb5hltiyzzdsrcyay3ffzugc5i

On the Positive Geometry of Conformal Field Theory [article]

Nima Arkani-Hamed, Yu-tin Huang, Shu-Heng Shao
2019 arXiv   pre-print
This lets us fully characterize the intersection U and X by a simple combinatorial rule, leading to a number of new exact statements about the spectrum and four-point function in any conformal field theory  ...  We study conformal blocks for the minimal SL(2,R) symmetry present in conformal field theories in all dimensions.  ...  Let us give a few examples of cyclic polytopes in two and three dimensions. For d = 2, any convex polygon is a cyclic polytope.  ... 
arXiv:1812.07739v2 fatcat:aj4ihqgo6re7vicrxlmtk2yozq

On smooth Gorenstein polytopes [article]

Benjamin Lorenz, Benjamin Nill
2013 arXiv   pre-print
These objects are of interest in combinatorial commutative algebra and enumerative combinatorics, and play a crucial role in Batyrev's and Borisov's computation of Hodge numbers of mirror-symmetric generic  ...  In this paper, we report on what is known about smooth Gorenstein polytopes, i.e., Gorenstein polytopes whose normal fan is unimodular.  ...  The complete intersection Calabi-Yau manifolds described in Theorem 2.3 are given by a so-called nef-partition, see [Bor93, BN08] .  ... 
arXiv:1303.2138v1 fatcat:hhdevs2ofzhcndseojg5astcoa

A new offspring of PALP [article]

Andreas P. Braun, Nils-Ole Walliser
2011 arXiv   pre-print
It is part of PALP, a package for analyzing lattice polytopes. Its main purpose is the construction and analysis of three--dimensional smooth Calabi--Yau hypersurfaces in toric varieties.  ...  Furthermore, it computes the intersection rings and characteristic classes of hypersurfaces.  ...  We would like to thank Harald Skarke for valuable advice in the completion phase of the program, and Christoph Mayrhofer for his comments and contributions to the source code at an early stage.  ... 
arXiv:1106.4529v1 fatcat:eg7rn2zs3nbrjauhxtahgfpp6u

Realization spaces of 4-polytopes are universal [article]

Jürgen Richter-Gebert, Günter M. Ziegler
1995 arXiv   pre-print
No similar universality result was previously known in any fixed dimension.  ...  This implies that the realization space of a 4-polytope can have the homotopy type of an arbitrary finite simplicial complex, and that all algebraic numbers are needed to realize all 4- polytopes.  ...  Thus, modulo projective equivalence, P (V ) contains a centrally symmetric (2n + 6)-gon whose slopes of opposite edges in any realization of P (V ) encode the coordinates of the corresponding point in  ... 
arXiv:math/9510217v1 fatcat:bxd3o2hdsrgonj6rhlo3mmt25m

Realization spaces of\\ 4-polytopes are universal

Jurgen Richter-Gebert, Guenter M. Ziegler
1995 Bulletin of the American Mathematical Society  
No similar universality result was previously known in any fixed dimension.  ...  This implies that the realization space of a 4-polytope can have the homotopy type of an arbitrary finite simplicial complex, and that all algebraic numbers are needed to realize all 4-polytopes.  ...  Thus, modulo projective equivalence, P (V ) contains a centrally symmetric (2n + 6)-gon whose slopes of opposite edges in any realization of P (V ) encode the coordinates of the corresponding point in  ... 
doi:10.1090/s0273-0979-1995-00604-x fatcat:ndxu3zrrunhivljsl75rmfijbu
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