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

2001
*
Lecture Notes in Computer Science
*

Our algorithms can be modified to count the number

doi:10.1007/3-540-44634-6_12
fatcat:jd2ynu3h4zef7k463fiavapazi
*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*. ...##
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Reporting intersecting pairs of convex polytopes in two and three dimensions

2002
*
Computational geometry
*

Our algorithms can be modified to count the number

doi:10.1016/s0925-7721(02)00049-4
fatcat:wl2swhgeubdfrnh3oyg7gm6kr4
*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*...##
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Splitting a complex of convex polytopes in any dimension

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. ...

##
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Conservative interpolation between volume meshes by local Galerkin projection

2011
*
Computer Methods in Applied Mechanics and Engineering
*

This paper presents an algorithm for the local implementation

doi:10.1016/j.cma.2010.07.015
fatcat:wdudtrxfaff5tg547vvww623ve
*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. ...
The algorithm uses a

doi:10.1145/199404.199437
dblp:conf/si3d/CohenLMP95
fatcat:4ehyercrijfchiyjsjdsufjcvu
*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. ...##
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Translational packing of arbitrary polytopes

2009
*
Computational geometry
*

Additional details are given for the

doi:10.1016/j.comgeo.2008.06.003
fatcat:if6hpdq2o5aelaa4bswxxw5kyy
*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. ...##
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Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

2013
*
Discrete & Computational Geometry
*

For any fixed d, we show how to compute the set

doi:10.1007/s00454-013-9503-3
fatcat:hpbcxdboi5f5fmrhweup63cbbi
*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. ...##
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Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space

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

doi:10.1145/1839778.1839780
dblp:conf/sma/MayerFH10
fatcat:lozk3gn3ovfebpy3sbpzbpfpyq
*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*. ...##
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Compact hyperbolic Coxeter five-dimensional polytopes with nine facets
[article]

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 ...

##
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Asymmetric Convex Intersection Testing

2018
*
ACM-SIAM Symposium on Discrete Algorithms
*

We consider asymmetric convex

doi:10.4230/oasics.sosa.2019.9
dblp:conf/soda/BarbaM19
fatcat:nb5hltiyzzdsrcyay3ffzugc5i
*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*. ...##
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On the Positive Geometry of Conformal Field Theory
[article]

2019
*
arXiv
*
pre-print

This lets us fully characterize the

arXiv:1812.07739v2
fatcat:aj4ihqgo6re7vicrxlmtk2yozq
*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*. ...##
###
On smooth Gorenstein polytopes
[article]

2013
*
arXiv
*
pre-print

These objects are

arXiv:1303.2138v1
fatcat:hhdevs2ofzhcndseojg5astcoa
*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] . ...##
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A new offspring of PALP
[article]

2011
*
arXiv
*
pre-print

It is part

arXiv:1106.4529v1
fatcat:eg7rn2zs3nbrjauhxtahgfpp6u
*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. ...##
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Realization spaces of 4-polytopes are universal
[article]

1995
*
arXiv
*
pre-print

No similar universality result was previously known

arXiv:math/9510217v1
fatcat:bxd3o2hdsrgonj6rhlo3mmt25m
*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*...##
###
Realization spaces of\\ 4-polytopes are universal

1995
*
Bulletin of the American Mathematical Society
*

No similar universality result was previously known

doi:10.1090/s0273-0979-1995-00604-x
fatcat:ndxu3zrrunhivljsl75rmfijbu
*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*...
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