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Capturing crossings: Convex hulls of segment and plane intersections

2008
*
Information Processing Letters
*

We give a simple O(n log n) algorithm to compute the

doi:10.1016/j.ipl.2008.03.024
fatcat:xx7j2gma35gzxew4rotn4fflxe
*convex**hull**of*the (possibly Θ(n 2 ))*intersection*points in an arrangement*of*n line*segments*in the*plane*. ... We also show an arrangement*of*dn*planes*in d-dimensions whose arrangement has Θ(n d−1 )*intersection*points on the*convex**hull*. ... Acknowledgments This research has been partially supported by grants from the National Science Foundation (CCF-0431030, CCF-0528209, CCA-0429901), Metron Aviation (NASA Ames subcontract),*and*NGA/Darpa ...##
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Density of Range Capturing Hypergraphs
[article]

2015
*
arXiv
*
pre-print

For a finite set X

arXiv:1404.1298v2
fatcat:ajejnqrzqrd3tc5usvexjcl4iy
*of*points in the*plane*, a set S in the*plane*,*and*a positive integer k, we say that a k-element subset Y*of*X is*captured*by S if there is a homothetic copy S'*of*S such that X∩ S' = ... A k-uniform S-*capturing*hypergraph H = H(X,S,k) has a vertex set X*and*a hyperedge set consisting*of*all k-element subsets*of*X*captured*by S. ... Acknowledgments We would like to thank Günter Rote*and*János Pach for helpful remarks*and*referring us first to [2]*and*later on to [5] . ...##
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Density of range capturing hypergraphs

2015
*
Journal of Computational Geometry
*

$\newcommand{\cH}{\mathcal{H}}$For a finite set $X$

doi:10.20382/jocg.v7i1a1
dblp:journals/jocg/AxenovichU16
fatcat:5drmjzmbpzca3ioi5isi37m4dm
*of*points in the*plane*, a set $S$ in the*plane*,*and*a positive integer $k$, we say that a $k$-element subset $Y$*of*$X$ is*captured*by $S$ if there is ... A $k$-uniform $S$-*capturing*hypergraph $\cH = \cH(X,S,k)$ has a vertex set $X$*and*a hyperedge set consisting*of*all $k$-element subsets*of*$X$*captured*by $S$. ... Acknowledgments We would like to thank Günter Rote*and*János Pach for helpful remarks*and*referring us to relevant literature. ...##
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Let Them Fall Where They May: Capture Regions of Curved Objects and Polyhedra

1997
*
The international journal of robotics research
*

Algorithms have been fully implemented for objects with smooth

doi:10.1177/027836499701600402
fatcat:pd2sfysfvngknnkwcikbxhf6cu
*and*polyhedral*convex**hulls*. ... When a three dimensional object is placed in contact with a supporting*plane*, gravitational forces move it to one*of*a nite set*of*stable poses. ... Je Wiegley provided the*convex**hull*software*and*polyhedral object models used in this paper. ...##
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Non-Crossing Frameworks with Non-Crossing Reciprocals

2004
*
Discrete & Computational Geometry
*

As in other recent papers where the interplay

doi:10.1007/s00454-004-1139-x
fatcat:che6rcal2jdyxjwbwuwnsfbh7q
*of*non-crossingness*and*rigidity*of*straight-line*plane*graphs is studied, pseudo-triangulations show up as objects*of*special interest. ... We study non-*crossing*frameworks in the*plane*for which the classical reciprocal on the dual graph is also non-*crossing*. ... Acknowledgements This research was initiated at the Workshop on Rigidity Theory*and*Scene Analysis organized by Ileana Streinu at the Bellairs Research Institute*of*McGill University in Barbados, Jan. ...##
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Partwise Cross-Parameterization via Nonregular Convex Hull Domains

2011
*
IEEE Transactions on Visualization and Computer Graphics
*

We first construct

doi:10.1109/tvcg.2010.231
pmid:21817171
fatcat:o7ovtrrk2bfppdqpecpweupfx4
*convex**hulls*for each part*of*the*segmented*model,*and*then adopt our*convex*-*hull**cross*-parameterization method to generate compatible meshes. ... Our method exploits properties*of*the*convex**hull*, e.g., good approximation ability*and*linear*convex*representation for interior vertices. ... .,*plane**and*sphere), the*convex**hull*provides a compact approximation for the*segmented*part. ...##
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-smooth branching surface construction from cross sections

2007
*
Computer-Aided Design
*

This paper proposes a framework for constructing G 1 surfaces that interpolate data points on parallel

doi:10.1016/j.cad.2007.05.004
fatcat:evz2uuzg2rd7rbst5umph4aj7y
*cross*sections, consisting*of*simple disjoined*and*non-nested contours, the number*of*which may vary ... The solution*of*the "one-to-many" branching problem, where one contour on the i-*plane*is to be connected to M contours on the (i + 1)*plane*, is based on combining skinning with trimming*and*hole filling ... Fig. 3 . 3 A set*of*M = 6 coplanar contours*and*their*convex**hull*. ...##
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The Rectilinear Crossing Number of K_10 is 62
[article]

2000
*
arXiv
*
pre-print

A drawing

arXiv:cs/0009023v1
fatcat:mkvihuhyifhzfewfqfsjqqcvni
*of*a graph G in the*plane*is said to be a rectilinear drawing*of*G if the edges are required to be line*segments*(as opposed to Jordan curves). We assume no three vertices are collinear. ... The rectilinear*crossing*number*of*G is the fewest number*of*edge*crossings*attainable over all rectilinear drawings*of*G. ...*Crossing*number problems are inherently rich*and*numerous,*and*have*captured*the attention*of*a diverse community*of*researchers. ...##
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The Rectilinear Crossing Number of $K_{10}$ is $62$

2001
*
Electronic Journal of Combinatorics
*

The rectilinear

doi:10.37236/1567
fatcat:4r5np3obqbfvfcrz2fe7wqkg2u
*crossing*number*of*a graph $G$ is the minimum number*of*edge*crossings*that can occur in any drawing*of*$G$ in which the edges are straight line*segments**and*no three vertices are collinear ... Acknowledgments*Crossing*number problems are easy to state but notoriously*and*profoundly difficult to solve. ... Throughout our investigations we discovered the inadequacy*of*simply searching the existing literature databases*and*subsequently reading papers. ...##
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Circles and crossing planar compact convex sets
[article]

2018
*
arXiv
*
pre-print

Let K_0 be a compact

arXiv:1802.06457v1
fatcat:weo7xggqjrhchdvio3iocu45dq
*convex*subset*of*the*plane*R^2,*and*assume that whenever K_1⊆ R^2 is congruent to K_0, then K_0*and*K_1 are not*crossing*in a natural sense due to L. Fejes-Tóth. A theorem*of*L. ... Our purpose is to describe the hierarchy among several variants*of*the new concepts*and*the old concept*of**crossing*. ... Hence the line*segment*[U L , U L ]*intersects*D = [A, B] at an inner point V . By the*convexity**of*L, we have that V ∈ L. ...##
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The density of iterated crossing points and a gap result for triangulations of finite point sets
[article]

2006
*
arXiv
*
pre-print

For a given point set P in R^2, we connect every pair

arXiv:cs/0601033v1
fatcat:h3jnb2yehfborbzs3r2uxoesma
*of*points in P by a line*segment**and*then add to P all those points where two such line*segments**cross*. ... For every pair a,b*of*vertices*of*G, we compare the shortest-path distance between a*and*b in G (with Euclidean edge lengths) to their actual distance in the*plane*. ... The limit set*of*line*intersections**of*a point set that contains four points, one*of*them in the interior*of*the*convex**hull**of*the others, is everywhere dense in the whole*plane*. ...##
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Circles and crossing planar compact convex sets

2019
*
Acta Scientarum Mathematicarum
*

Let K0 be a compact

doi:10.14232/actasm-018-522-2
fatcat:7guffjiflncvpaol4l766yepte
*convex*subset*of*the*plane*R 2 ,*and*assume that whenever K1 ⊆ R 2 is congruent to K0, then K0*and*K1 are not*crossing*in a natural sense due to L. Fejes-Tóth. A theorem*of*L. ... Our purpose is to describe the hierarchy among several variants*of*the new concepts*and*the old concept*of**crossing*. ... Hence the line*segment*[U L , U ′ L ]*intersects*D = [A, B]at an inner point V . By the*convexity**of*L, we have that V ∈ L. ...##
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A Markerless Motion Capture System to Study Musculoskeletal Biomechanics: Visual Hull and Simulated Annealing Approach

2006
*
Annals of Biomedical Engineering
*

The described method is based on visual

doi:10.1007/s10439-006-9122-8
pmid:16783657
fatcat:lup4mmkytbbnlippwlqdajdl4m
*hull*reconstruction*and*an a priori model*of*the subject. ... A custom version*of*adapted fast simulated annealing has been developed to match the model to the visual*hull*. ... Their*intersection*generates the visual*hull*, a locally*convex*over-approximation*of*the volume occupied by the subject's body. ...##
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Geometric analysis of cross-linkability for protein fold discrimination

2004
*
Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing
*

This paper presents a suite

pmid:14992524
fatcat:pk2ixt6u4fa6lav5nopyri2u7a
*of*novel lower-*and*upper-bounding algorithms for analyzing the distance between surface*cross*-link sites*and*thereby validating predicted models against experimental*cross*-linking ... Protein structure provides insight into the evolutionary origins, functions,*and*mechanisms*of*proteins. ... Thanks to Mike Stoppelman, Xiaoduan Ye,*and*other members*of*our labs for helpful discussions*and*related work. ...##
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GEOMETRIC ANALYSIS OF CROSS-LINKABILITY FOR PROTEIN FOLD DISCRIMINATION

2003
*
Biocomputing 2004
*

This paper presents a suite

doi:10.1142/9789812704856_0042
fatcat:s7puzvflsja7be6ou275yfotq4
*of*novel lower-*and*upper-bounding algorithms for analyzing the distance between surface*cross*-link sites*and*thereby validating predicted models against experimental*cross*-linking ... Protein structure provides insight into the evolutionary origins, functions,*and*mechanisms*of*proteins. ... Thanks to Mike Stoppelman, Xiaoduan Ye,*and*other members*of*our labs for helpful discussions*and*related work. ...
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