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Capturing crossings: Convex hulls of segment and plane intersections

Esther M. Arkin, Joseph S.B. Mitchell, Jack Snoeyink
2008 Information Processing Letters  
We give a simple O(n log n) algorithm to compute the 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  ... 
doi:10.1016/j.ipl.2008.03.024 fatcat:xx7j2gma35gzxew4rotn4fflxe

Density of Range Capturing Hypergraphs [article]

Maria Axenovich, Torsten Ueckerdt
2015 arXiv   pre-print
For a finite set X 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] .  ... 
arXiv:1404.1298v2 fatcat:ajejnqrzqrd3tc5usvexjcl4iy

Density of range capturing hypergraphs

Maria Axenovich, Torsten Ueckerdt
2015 Journal of Computational Geometry  
$\newcommand{\cH}{\mathcal{H}}$For a finite set $X$ 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.  ... 
doi:10.20382/jocg.v7i1a1 dblp:journals/jocg/AxenovichU16 fatcat:5drmjzmbpzca3ioi5isi37m4dm

Let Them Fall Where They May: Capture Regions of Curved Objects and Polyhedra

David J. Kriegman
1997 The international journal of robotics research  
Algorithms have been fully implemented for objects with smooth 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.  ... 
doi:10.1177/027836499701600402 fatcat:pd2sfysfvngknnkwcikbxhf6cu

Non-Crossing Frameworks with Non-Crossing Reciprocals

David Orden, G�nter Rote, Francisco Santos, Brigitte Servatius, Herman Servatius, Walter Whiteley
2004 Discrete & Computational Geometry  
As in other recent papers where the interplay 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.  ... 
doi:10.1007/s00454-004-1139-x fatcat:che6rcal2jdyxjwbwuwnsfbh7q

Partwise Cross-Parameterization via Nonregular Convex Hull Domains

Huai-Yu Wu, Chunhong Pan, Hongbin Zha, Qing Yang, Songde Ma
2011 IEEE Transactions on Visualization and Computer Graphics  
We first construct 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.  ... 
doi:10.1109/tvcg.2010.231 pmid:21817171 fatcat:o7ovtrrk2bfppdqpecpweupfx4

-smooth branching surface construction from cross sections

N.C. Gabrielides, A.I. Ginnis, P.D. Kaklis, M.I. Karavelas
2007 Computer-Aided Design  
This paper proposes a framework for constructing G 1 surfaces that interpolate data points on parallel 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.  ... 
doi:10.1016/j.cad.2007.05.004 fatcat:evz2uuzg2rd7rbst5umph4aj7y

The Rectilinear Crossing Number of K_10 is 62 [article]

Alex Brodsky, Stephane Durocher, Ellen Gethner
2000 arXiv   pre-print
A drawing 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.  ... 
arXiv:cs/0009023v1 fatcat:mkvihuhyifhzfewfqfsjqqcvni

The Rectilinear Crossing Number of $K_{10}$ is $62$

Alex Brodsky, Stephane Durocher, Ellen Gethner
2001 Electronic Journal of Combinatorics  
The rectilinear 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.  ... 
doi:10.37236/1567 fatcat:4r5np3obqbfvfcrz2fe7wqkg2u

Circles and crossing planar compact convex sets [article]

Gábor Czédli
2018 arXiv   pre-print
Let K_0 be a compact 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.  ... 
arXiv:1802.06457v1 fatcat:weo7xggqjrhchdvio3iocu45dq

The density of iterated crossing points and a gap result for triangulations of finite point sets [article]

Rolf Klein, Martin Kutz
2006 arXiv   pre-print
For a given point set P in R^2, we connect every pair 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.  ... 
arXiv:cs/0601033v1 fatcat:h3jnb2yehfborbzs3r2uxoesma

Circles and crossing planar compact convex sets

Gábor Czédli
2019 Acta Scientarum Mathematicarum  
Let K0 be a compact 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.  ... 
doi:10.14232/actasm-018-522-2 fatcat:7guffjiflncvpaol4l766yepte

A Markerless Motion Capture System to Study Musculoskeletal Biomechanics: Visual Hull and Simulated Annealing Approach

S. Corazza, L. Mündermann, A. M. Chaudhari, T. Demattio, C. Cobelli, T. P. Andriacchi
2006 Annals of Biomedical Engineering  
The described method is based on visual 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.  ... 
doi:10.1007/s10439-006-9122-8 pmid:16783657 fatcat:lup4mmkytbbnlippwlqdajdl4m

Geometric analysis of cross-linkability for protein fold discrimination

S Potluri, A A Khan, A Kuzminykh, J M Bujnicki, A M Friedman, C Bailey-Kellogg
2004 Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing  
This paper presents a suite 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.  ... 
pmid:14992524 fatcat:pk2ixt6u4fa6lav5nopyri2u7a

GEOMETRIC ANALYSIS OF CROSS-LINKABILITY FOR PROTEIN FOLD DISCRIMINATION

S. POTLURI, A.A. KHAN, A. KUZMINYKH, J.M. BUJNICKI, A.M. FRIEDMAN, C. BAILEY-KELLOGG
2003 Biocomputing 2004  
This paper presents a suite 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.  ... 
doi:10.1142/9789812704856_0042 fatcat:s7puzvflsja7be6ou275yfotq4
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