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On lines avoiding unit balls in three dimensions

Pankaj K. Agarwal, Boris Aronov, Vladlen Koltun, Micha Sharir
2004 Proceedings of the twentieth annual symposium on Computational geometry - SCG '04  
Let B be a set of n unit balls in R 3 . We show that the combinatorial complexity of the space of lines in R 3 that avoid all the balls of B is O(n 3+ε ), for any ε > 0.  ...  This result has connections to problems in visibility, ray shooting, motion planning and geometric optimization.  ...  Acknowledgments The authors would like to express their appreciation of the city of Heidelberg, whose magic ambiance was the fertile soil in which, in March 2003, the initial seeds of this work were planted  ... 
doi:10.1145/997817.997826 dblp:conf/compgeom/AgarwalAKS04 fatcat:in76c36xrrcwrioqwmnf7okm6m

Intrinsically anomalous self-similarity of randomly folded matter

Alexander S. Balankin, Rolando Cortes Montes de Oca, Didier Samayoa Ochoa
2007 Physical Review E  
dimension D Ͻ D l .  ...  of crumpling creases, rather than by the condition of self-avoidance.  ...  units; ͑c͒ ⌳ / R versus K in arbitrary units; and ͑d͒ ⌫ / R versus K 3 , in arbitrary units, for balls folded from three different papers: carbon ͑᭺͒, biblia ͑b͒, and albanene ͑छ͒.  ... 
doi:10.1103/physreve.76.032101 pmid:17930290 fatcat:higj3wvaondynhd6jzpqr4wiom

Page 174 of POWER Vol. 86, Issue 10 [page]

1942 POWER  
Specification sheets, 9, cover company’s line of diesel-generator sets, engines and power units in three sizes, 2, 4 and 6 cylinders.  ...  They are essential to avoid undue strain on fittings and expansion joints; good assurance against future line failure. Know your actual steam consumption for heating or process purposes.  ... 

A unified approach to the deformation of simplicial and non-simplicial meshes in two and three dimensions with guaranteed validity

Nazmiye Acikgoz, Carlo L. Bottasso
2007 Computers & structures  
Specifically, an invalid element is formed when a mesh vertex leaves its ball, i.e. the polyhedral cavity formed by all triangular faces (in three dimensions) or edges (in two dimensions) that are edge-connected  ...  Next, we show that, in all cases above, collapse of elements is avoided by connecting with a spring each vertex in the grid with its normal projection on the ball boundaries.  ...  The method is based on a network of linear face-vertex springs in three dimensions or linear edge-vertex springs in two dimensions.  ... 
doi:10.1016/j.compstruc.2006.11.009 fatcat:75b4unqqvnao7huopnk3f2ghi4

Higher-Dimensional Analogues of the Map Coloring Problem

Bhaskar Bagchi, Basudeb Datta
2013 The American mathematical monthly  
After a brief discussion of the history of the problem, we propose a generalization of the map colouring problem to higher dimensions.  ...  The same observation goes for all higher dimensions since the three dimensional space embeds in all Euclidean spaces of higher dimensions.  ...  In these dimensions one may argue (exactly as we have done in the case d = 2) that χ(d) ≤ κ(d), a slight improvement ! In our familiar three dimensional space, we have 5 ≤ χ(3) ≤ 13.  ... 
doi:10.4169/amer.math.monthly.120.08.733 fatcat:k3rp65fsgrarzcqckqfmcc5ugm

Higher dimensional analogues of the map colouring problem [article]

Bhaskar Bagchi, Basudeb Datta
2012 arXiv   pre-print
After a brief discussion of the history of the problem, we propose a generalization of the map colouring problem to higher dimensions.  ...  The same observation goes for all higher dimensions since the three dimensional space embeds in all Euclidean spaces of higher dimensions.  ...  In these dimensions one may argue (exactly as we have done in the case d = 2) that χ(d) ≤ κ(d), a slight improvement ! In our familiar three dimensional space, we have 5 ≤ χ(3) ≤ 13.  ... 
arXiv:1202.0153v1 fatcat:eopmhysttrgppjvbg6xqwlpatu

Some Discrete Properties of the Space of Line Transversals to Disjoint Balls [chapter]

Xavier Goaoc
2009 IMA Volumes in Mathematics and its Applications  
We review recent progress in the special case of disjoint Euclidean balls in R d , more precisely the inter-related notions of cone of directions, geometric permutations and Helly-type theorems, and discuss  ...  Attempts to generalize Helly's theorem to sets of lines intersecting convex sets led to a series of results relating the geometry of a family of sets in R d to the structure of the space of lines intersecting  ...  in three dimensions [1] ; if the dimension is part of the input, deciding if (intersecting) unit balls have a transversal is NP-hard [59] .  ... 
doi:10.1007/978-1-4419-0999-2_3 fatcat:snteyqtjojh6rjbeui6r4ih65m

On the Helly Number for Hyperplane Transversals to Unit Balls

B. Aronov, J. E. Goodman, R. Pollack, R. Wenger
2000 Discrete & Computational Geometry  
of unit balls in d-dimensional Euclidean space.  ...  Helly number for hyperplane transversals to suitably separated families of unit balls in R d .  ...  In [4] Danzer also considered briefly two variants of the problem in higher dimensions: one for line transversals of pairwise disjoint unit balls (which we will not touch on here), the other for hyperplane  ... 
doi:10.1007/s004540010024 fatcat:w4s4o45wubgqhajc5dpvdfq3em

The ball-vertex method: a new simple spring analogy method for unstructured dynamic meshes

Carlo L. Bottasso, Davide Detomi, Roberto Serra
2005 Computer Methods in Applied Mechanics and Engineering  
This is here achieved by confining each mesh vertex to its ball, through linear springs that are attached to the vertex and to its projections on the ball entities.  ...  The proposed procedures are compared with a revisited version of the torsional spring method with the help of two and three-dimensional example problems.  ...  problems, especially in three spatial dimensions.  ... 
doi:10.1016/j.cma.2004.08.014 fatcat:q5ebhlz27bb5vnhnyxlv2nojyy

Comparing 3D foot scanning with conventional measurement methods

Yu-Chi Lee, Gloria Lin, Mao-Jiun J Wang
2014 Journal of Foot and Ankle Research  
Foot dimension information on different user groups is important for footwear design and clinical applications.  ...  Two-way ANOVA was performed to evaluate the sex and method effect on the measured foot dimensions.  ...  measurement methods (as shown in Figure 2 2 The sex differences in six foot dimensions among measuring methods (unit in mm).  ... 
doi:10.1186/s13047-014-0044-7 pmid:25364389 pmcid:PMC4215017 fatcat:v6urgmnge5bjzozmlaukotm5q4

Comparing 3D foot scanning with conventional measurement methods

Yu-Chi Lee, Gloria Lin, Mao-Jiun J Wang
2014 Journal of Foot and Ankle Research  
Foot dimension information on different user groups is important for footwear design and clinical applications.  ...  Two-way ANOVA was performed to evaluate the sex and method effect on the measured foot dimensions.  ...  measurement methods (as shown in Figure 2 2 The sex differences in six foot dimensions among measuring methods (unit in mm).  ... 
doi:10.1186/preaccept-7889828871251676 fatcat:taflszhrjng3tk5z6n23ldb63e

Lines Avoiding Unit Balls in Three Dimensions

Pankaj K. Agarwal, Boris Aronov, Vladlen Koltun, Micha Sharir
2005 Discrete & Computational Geometry  
Let B be a set of n unit balls in R 3 . We show that the combinatorial complexity of the space of lines in R 3 that avoid all the balls of B is O(n 3+ε ), for any ε > 0.  ...  This result has connections to problems in visibility, ray shooting, motion planning, and geometric optimization.  ...  Acknowledgments The authors express their appreciation of the city of Heidelberg, whose magic ambiance was the fertile soil in which, in March 2003, the initial seeds of this work were planted and began  ... 
doi:10.1007/s00454-005-1166-2 fatcat:ksnsnus335b37fbxijmcslyala

Local intrinsic dimensionality estimators based on concentration of measure [article]

Jonathan Bac, Andrei Zinovyev
2020 arXiv   pre-print
In this paper, we introduce new local estimators of ID based on linear separability of multi-dimensional data point clouds, which is one of the manifestations of concentration of measure.  ...  Intrinsic dimensionality (ID) is one of the most fundamental characteristics of multi-dimensional data point clouds.  ...  Other estimator names are referenced in the Introduction; Glocal and global pointwise ID estimates computed on a synthetic dataset containing 10 unit balls of different dimensions (2D to 11D, each sampled  ... 
arXiv:2001.11739v3 fatcat:7zrnugdtundy7p74mw2enxme6q

Open Sets Avoiding Integral Distances

Sascha Kurz, Valery Mishkin
2013 Discrete & Computational Geometry  
We determine the lower and upper bounds for the volumes of the sets in terms of the number of their connected components and dimension, and also give some exact values.  ...  Our problem can be viewed as a kind of inverse to known problems on sets with pairwise rational or integral distances.  ...  Steprans, Peter Biryukov, Robert Connelly, and Andrey Verevkin for valuable comments and discussions, Tobias Kreisel for producing several graphics, Thomas Kriecherbauer and Benoit Cloitre for some remarks on  ... 
doi:10.1007/s00454-013-9508-y fatcat:mjgh2l7ggnep5ofd4jjkxbsm7i

Fractal features of a crumpling network in randomly folded thin matter and mechanics of sheet crushing

Alexander S. Balankin, Antonio Horta Rangel, Gregorio García Pérez, Felipe Gayosso Martinez, Hugo Sanchez Chavez, Claudia L. Martínez-González
2013 Physical Review E  
The effect of compaction geometry on the crushing mechanics is revealed.  ...  We study the static and dynamic properties of networks of crumpled creases formed in hand crushed sheets of paper.  ...  Crumpling network Balls folded from paper sheet On unfolded flat sheet In sheet folded in ball Chemical dimension 2 1.6 ± 0.1 (≈5/3) Global mass dimension Dependent on the folding Not defined  ... 
doi:10.1103/physreve.87.052806 pmid:23767583 fatcat:k7upc4kmcnf2bgqvt6ttjuiljy
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