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Finding a best approximation pair of points for two polyhedra

Ron Aharoni, Yair Censor, Zilin Jiang
2018 Computational optimization and applications  
Given 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.  ... 
doi:10.1007/s10589-018-0021-3 fatcat:dqpv4qttijeuzfweflp2nao7eq

On the complexity of approximating and illuminating three-dimensional convex polyhedra [chapter]

Gautam Das, Michael T. Goodrich
1995 Lecture Notes in Computer Science  
One 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.  ... 
doi:10.1007/3-540-60220-8_52 fatcat:ztihkx3lkbhq3czsuliwe3ub5i

A comparison of two fast algorithms for computing the distance between convex polyhedra

S. Cameron
1997 IEEE Transactions on Robotics and Automation  
Abstract| The problem 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.  ... 
doi:10.1109/70.650170 fatcat:n6d3rsat5nbzpo5ao7hr7a3sdq

An iterative algorithm for finding a nearest pair of points in two convex subsets of Rn

B. Llanas, M. Fernandez de Sevilla, V. Feliu
2000 Computers and Mathematics with Applications  
We present an algorithm 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.  ... 
doi:10.1016/s0898-1221(00)85008-7 fatcat:sivbnlitarftva33giclaan4yy

The Roundest Polyhedra with Symmetry Constraints

András Lengyel, Zsolt Gáspár, Tibor Tarnai
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.  ... 
doi:10.3390/sym9030041 fatcat:u2stjpbq3zhzxmjkrxxigyf32a

Voronoi Polyhedra Analysis of Optimized Arterial Tree Models

Rudolf Karch, Friederike Neumann, Martin Neumann, Paul Szawlowski, Wolfgang Schreiner
2003 Annals of Biomedical Engineering  
The distributions 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.  ... 
doi:10.1114/1.1566444 pmid:12757199 fatcat:jl3iq6hplrcm5elv2epnhdwptm

Separation and approximation of polyhedral objects

Joseph S.B. Mitchell, Subhash Suri
1995 Computational geometry  
In three dimensions, the problem is NP-complete even 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  ... 
doi:10.1016/0925-7721(95)00006-u fatcat:wzf7ohyjerdmvjsxrdwkblb2ny

H-Walk

Leonidas J. Guibas, David Hsu, Li Zhang
1999 Proceedings of the fifteenth annual symposium on Computational geometry - SCG '99  
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.  ... 
doi:10.1145/304893.304979 dblp:conf/compgeom/GuibasHZ99 fatcat:wkw554g4bfgyjkyr2yrl2gb4gy

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.  ... 

Volume matching with application in medical treatment planning

Yam Ki Cheung, Ovidiu Daescu, Lech Papiez
2011 Proceedings of the 2nd ACM Conference on Bioinformatics, Computational Biology and Biomedicine - BCB '11  
The computation 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.  ... 
doi:10.1145/2147805.2147878 dblp:conf/bcb/CheungDP11 fatcat:cs43ais7wjht3er6cubxobkbgi

Volume matching with application in medical treatment planning

Yam Ki Cheung, Ovidiu Daescu, Steven Kirtzic, Lech Papiez
2012 2012 IEEE 2nd International Conference on Computational Advances in Bio and medical Sciences (ICCABS)  
The computation 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.  ... 
doi:10.1109/iccabs.2012.6182630 dblp:conf/iccabs/CheungDKP12 fatcat:qlamxziwbzbfhi4a4oy3umzmbe

A hierarchical method for real-time distance computation among moving convex bodies

Leonidas J. Guibas, David Hsu, Li Zhang
2000 Computational geometry  
by subsets 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.  ... 
doi:10.1016/s0925-7721(99)00042-5 fatcat:ylgvqqxjgbarjbly4lrtek6jly

Dynamic Minkowski sum of convex shapes

Evan Behar, Jyh-Ming Lien
2011 2011 IEEE International Conference on Robotics and Automation  
We propose 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.  ... 
doi:10.1109/icra.2011.5979992 dblp:conf/icra/BeharL11 fatcat:jsl6wfxdsrfynnksdgyaewkbda

Intersecting dilated convex polyhedra method for modeling complex particles in discrete element method

Ben Nye, Anton V. Kulchitsky, Jerome B. Johnson
2014 International journal for numerical and analytical methods in geomechanics (Print)  
This paper describes 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.  ... 
doi:10.1002/nag.2299 pmid:26300584 pmcid:PMC4540157 fatcat:dbfdgxmspbb7rbupnvyveuit44

Proximity Queries Between Convex Objects: An Interior Point Approach for Implicit Surfaces

Nilanjan Chakraborty, Jufeng Peng, Srinivas Akella, John E. Mitchell
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.  ... 
doi:10.1109/tro.2007.914851 fatcat:ilcb2frqzfau7mdqbspk744mze
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