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Finding a best approximation pair of points for two polyhedra

2018
*
Computational optimization and applications
*

Given

doi:10.1007/s10589-018-0021-3
fatcat:dqpv4qttijeuzfweflp2nao7eq
*two*disjoint convex*polyhedra*, we look*for**a**best**approximation**pair*relative to them, i.e.,*a**pair**of**points*, one in each polyhedron, attaining the minimum distance between the sets. ... Cheney and Goldstein showed that alternating projections onto the*two*sets, starting from an arbitrary*point*, generate*a*sequence whose*two*interlaced subsequences converge to*a**best**approximation**pair*... Acknowledgements We thank Yehuda Zur*for*Matlab programming work at the early stages*of*our research. ...##
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On the complexity of approximating and illuminating three-dimensional convex polyhedra
[chapter]

1995
*
Lecture Notes in Computer Science
*

One

doi:10.1007/3-540-60220-8_52
fatcat:ztihkx3lkbhq3czsuliwe3ub5i
*of*the techniques we employ is*a*linear-time method*for*realizing*a*planar 3-connected triangulation as*a*convex polyhedron. ... We show that several well-known computational geometry problems involving 3-dimensional convex*polyhedra*are NP-hard or NP-complete. ... Acknowledgements We would like to thank Marek Chrobak*for*his encouragement and his diligence in*pointing*out places in previous versions*of*this paper that needed more details. ...##
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A comparison of two fast algorithms for computing the distance between convex polyhedra

1997
*
IEEE Transactions on Robotics and Automation
*

Abstract| The problem

doi:10.1109/70.650170
fatcat:n6d3rsat5nbzpo5ao7hr7a3sdq
*of*tracking the distance between*two*convex*polyhedra*is nding applications in many areas*of*robotics. ... The algorithm*of*Lin and Canny is*a*well-known fast solution to this problem, but by recasting the algorithms into con guration space we show that*a*minor modi cation to the earlier algorithm*of*Gilbert ... My thanks are also due to many other researchers*for*discussing this topic, and in particular to Chong Jin Ong and Elmer Gilbert*for*details*of*the workings*of*GJK. ...##
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An iterative algorithm for finding a nearest pair of points in two convex subsets of Rn

2000
*
Computers and Mathematics with Applications
*

We present an algorithm

doi:10.1016/s0898-1221(00)85008-7
fatcat:sivbnlitarftva33giclaan4yy
*for**finding**a*nearest*pair**of**points*in*two*convex sets*of*R n, and therefore, their distance. ... We introduce such*a*procedure*for*convex*polyhedra*. This algorithm effects*a*local search in the faces using visibility as*a*guide*for**finding*the global minimum. ... In Section 2*of*this article, we give the theoretic foundation*of*an iterative algorithm*for**finding**a*nearest*pair**of**points*in*two*convex, bounded, and closed subsets*of*R n. ...##
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The Roundest Polyhedra with Symmetry Constraints

2017
*
Symmetry
*

*A*new numerical optimization method developed previously by the authors has been applied to optimize

*polyhedra*to

*best*

*approximate*

*a*sphere if tetrahedral, octahedral, or icosahedral symmetry constraints ... The solution

*of*this problem represents the closest

*approximation*

*of*the sphere, i.e., the roundest

*polyhedra*. ... Conflicts

*of*Interest: The authors declare no conflict

*of*interest. ...

##
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Voronoi Polyhedra Analysis of Optimized Arterial Tree Models

2003
*
Annals of Biomedical Engineering
*

The distributions

doi:10.1114/1.1566444
pmid:12757199
fatcat:jl3iq6hplrcm5elv2epnhdwptm
*of*N f , V, and S*of*the CCO models are reasonably well*approximated*by*two*-parameter gamma distributions. ... Topological and metric properties*of*Voronoi*polyhedra*͑VP͒ generated by the distal end*points**of*terminal segments in arterial tree models grown by the method*of*constrained constructive optimization ͑ ... ACKNOWLEDGMENTS The authors gratefully acknowledge the valuable advice from the editor regarding the literature and the critical comments provided by the anonymous reviewers*of*this study. ...##
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Separation and approximation of polyhedral objects

1995
*
Computational geometry
*

In three dimensions, the problem is NP-complete even

doi:10.1016/0925-7721(95)00006-u
fatcat:wzf7ohyjerdmvjsxrdwkblb2ny
*for**two*. 0925-7721/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO925-7721(95)00006-2 96 ... Given*a*family*of*disjoint polygons P" P" . . . , Pk in the plane, and an integer parameter m, it is AT-complete to decide if the Pi's can be pairwise separated by*a*polygonal family with at most m edges ... Let pi E Pi and pi E Pj denote*two**points*that form*a*closest*pair**for*Pi and Pj: d( Pi 7 Pj) = min{ 4 x, The top view*of*our lowe I I bound construction*for*separating k convex*polyhedra*in three dimensions ...*For*convex polygons, we prove that H-Walk improves on the classic Lin-Canny and Dobkin-Kirkpatrick algorithms. We have implemented H-Walk

*for*moving convex

*polyhedra*in three dimensions. ... This paper presents the Hierarchical Walk, or H-Walk algorithm, which maintains the distance between

*two*moving convex bodies by exploiting both motion coherence and hierarchical representations. ... Acknowledgment We would like to thank Brian Mirtich

*for*helpful discussion

*of*the collision detection problem and

*for*providing us an implementation

*for*the V-Clip algorithm. ...

##
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Page 5494 of Mathematical Reviews Vol. , Issue 2003g
[page]

2003
*
Mathematical Reviews
*

The

*approximating*polytopes achieve the*best*possible general order*of*precision in the sense*of*volume-difference. ... The presented data structure contains*a*sequence*of*O(log) stretch factors, one from each subsequence. In addition,*a*sequence*of*O(logn) dis- tances between*pairs**of**points*is stored. ...##
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Volume matching with application in medical treatment planning

2011
*
Proceedings of the 2nd ACM Conference on Bioinformatics, Computational Biology and Biomedicine - BCB '11
*

The computation

doi:10.1145/2147805.2147878
dblp:conf/bcb/CheungDP11
fatcat:cs43ais7wjht3er6cubxobkbgi
*of*seeds placement is*a*difficult problem that requires*a*significant amount*of*time to solve. ... In this paper we address the problem*of*matching an input object (tumor) against*a*repository*of*model objects associated with known treatment plans such that the overlap volume is maximized. ... In 3-dimensions,*for**find*the maximum overlap volume between*two*given convex*polyhedra*P and Q, Ahn et al. ...##
###
Volume matching with application in medical treatment planning

2012
*
2012 IEEE 2nd International Conference on Computational Advances in Bio and medical Sciences (ICCABS)
*

The computation

doi:10.1109/iccabs.2012.6182630
dblp:conf/iccabs/CheungDKP12
fatcat:qlamxziwbzbfhi4a4oy3umzmbe
*of*seeds placement is*a*difficult problem that requires*a*significant amount*of*time to solve. ... In this paper we address the problem*of*matching an input object (tumor) against*a*repository*of*model objects associated with known treatment plans such that the overlap volume is maximized. ... In 3-dimensions,*for**find*the maximum overlap volume between*two*given convex*polyhedra*P and Q, Ahn et al. ...##
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A hierarchical method for real-time distance computation among moving convex bodies

2000
*
Computational geometry
*

by subsets

doi:10.1016/s0925-7721(99)00042-5
fatcat:ylgvqqxjgbarjbly4lrtek6jly
*of*these*points*[7] . ... Convexity guarantees that we can determine locally whether*a**pair**of*features is the closest*pair*, and if not,*a*neighboring*pair*that is closer. ... Acknowledgements We would like to thank Brian Mirtich*for*helpful discussion*of*the collision detection problem and*for*providing us an implementation*for*the V-Clip algorithm. ...##
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Dynamic Minkowski sum of convex shapes

2011
*
2011 IEEE International Conference on Robotics and Automation
*

We propose

doi:10.1109/icra.2011.5979992
dblp:conf/icra/BeharL11
fatcat:jsl6wfxdsrfynnksdgyaewkbda
*a*method, called DYMSUM, that can efficiently update the Minkowski sums*of*rotating convex*polyhedra*. ... We show that DYMSUM is significantly more efficient than the traditional approach, in particular when the size*of*the input*polyhedra*are large and when the rotation is small between frames. ... In order to*find*the compatible*pair*, we*find*the witness*of*the incompatibility e, and replace v with the other end*point*v = v*of*e, and repeat this until f and v become compatible. ...##
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Intersecting dilated convex polyhedra method for modeling complex particles in discrete element method

2014
*
International journal for numerical and analytical methods in geomechanics (Print)
*

This paper describes

doi:10.1002/nag.2299
pmid:26300584
pmcid:PMC4540157
fatcat:dbfdgxmspbb7rbupnvyveuit44
*a*new method*for*representing concave polyhedral particles in*a*discrete element method as unions*of*convex dilated*polyhedra*. ... spheres, tetrahedra, cubes, unions*of**two*octahedra (concave), and*a*model*of**a*computer tomography scan*of**a*lunar simulant GRC-3 particle. ... The authors would like to thank Paul Duvoy*for*performing*a*computer tomography 3D scan*of*lunar regolith simulant particles. ...##
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Proximity Queries Between Convex Objects: An Interior Point Approach for Implicit Surfaces

2008
*
IEEE Transactions on robotics
*

*For*the case

*of*

*polyhedra*and quadrics, we establish

*a*theoretical time complexity

*of*O(n 1.5 ), where n is the number

*of*constraints. ...

*For*

*polyhedra*and quadrics, we establish that the computational complexity

*of*this problem is O(n 1.5 ). ... Thanks to Richard Waltz

*for*help with KNITRO, Buck Clay

*for*graphics software, and Jeff Trinkle, Steve Berard, Binh Nguyen, and Frank Luk

*for*useful discussions. ...

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