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Exact Computation of the Voronoi Diagram of Spheres in 3D, Its Topology and Its Geometric Invariants

Francois Anton, Darka Mioc, Marcelo Santos
2011 2011 Eighth International Symposium on Voronoi Diagrams in Science and Engineering  
of this methodology to get all geometrical invariants that intervene in this problem and the exact computation of the Delaunay graph and the Voronoi diagram of spheres.  ...  To the best of our knowledge, there does not exist a comprehensive treatment of the exact computation with geometrical invariants of the Delaunay graph and the Voronoi diagram of spheres.  ...  diagram of 3D balls and its computation via tracing edges [17] and the Euclidean Voronoi diagrams of 3D spheres and applications to protein structure analysis [19] .  ... 
doi:10.1109/isvd.2011.16 dblp:conf/isvd/AntonMS11 fatcat:5amrfmprrnbctggn2ayoulz3ai

Skeletonisation Approaches for Determining Paths in an Image: A Review

V. Sangeetha, V. Vaithiyanathan, R. Sivagami, K. Divyalakshmi, K. Joseph Abraham Sundar, M. Ifjaz Ahmed
2015 Indian Journal of Science and Technology  
This approach has great applications in the fields of motion planning, object representation, tracking, information retrieval, topology representation, sensor distribution and location, facility location  ...  Quantitative measures for validating the skeleton are also discussed. Finally, the applications of Skeletonisation in various domains are elaborated.  ...  But Voronoi diagrams retain the topological and the geometrical nature of the image 12,13 .  ... 
doi:10.17485/ijst/2015/v8i35/86782 fatcat:ogs27auwgnhpvps3ehgc2cyla4

Developing performance metrics for a design engineering department

R.K. Buchheim
2000 IEEE transactions on engineering management  
Note that if the inscribed sphere is empty then its center is a vertex of Voronoi diagram.4.7 Exact Computation MethodAlgorithm in computationai geometry are usually designed under the assumption that  ...  The i=l geometric interpretation of the power distance is that it is the square of the distance between a point x and a sphere P computed along a line tangent to the sphere P (see Figure 3 .1. 1 31 Geometrical  ... 
doi:10.1109/17.865900 fatcat:vyj24odqznenbdtvfbf6gzacxi

Voronoi diagrams in quasi-2D hard sphere systems

David Ormrod Morley, Mark Wilson
2020 Journal of Statistical Mechanics: Theory and Experiment  
Significantly, we demonstrate that using an unweighted (no dependence on sphere size) two-dimensional Voronoi construction (in which the sphere centres are simply projected onto a single plane) is topologically  ...  Key network and Voronoi cell properties (the fraction of six-membered rings, assortativity and cell height) and are analysed as a function of separation from the basal plane and the limits discussed.  ...  Acknowledgments We are grateful for support from the EPSRC Centre for Doctoral Training in Theory and Modelling in Chemical Sciences (TMCS), under grant EP/L015722/1, and from the AWE.  ... 
doi:10.1088/1742-5468/aba7af fatcat:y5sh6etiqzhkfmevtvbllzsi6e

Matching fluid simulation elements to surface geometry and topology

Tyson Brochu, Christopher Batty, Robert Bridson
2010 ACM Transactions on Graphics  
Constructing the simulation mesh in this way allows us to place samples anywhere in the computational domain; we exploit this by choosing samples that accurately capture the geometry and topology of the  ...  Abstract We introduce an Eulerian liquid simulation framework based on the Voronoi diagram of a potentially unorganized collection of pressure samples.  ...  Acknowledgements This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.  ... 
doi:10.1145/1833351.1778784 fatcat:xnzumwxerzfxvf2btjxfkk3x24

Matching fluid simulation elements to surface geometry and topology

Tyson Brochu, Christopher Batty, Robert Bridson
2010 ACM SIGGRAPH 2010 papers on - SIGGRAPH '10  
Constructing the simulation mesh in this way allows us to place samples anywhere in the computational domain; we exploit this by choosing samples that accurately capture the geometry and topology of the  ...  Abstract We introduce an Eulerian liquid simulation framework based on the Voronoi diagram of a potentially unorganized collection of pressure samples.  ...  Acknowledgements This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.  ... 
doi:10.1145/1833349.1778784 fatcat:2w32f4nkofhbdk25jx7bh7mvw4

Matching fluid simulation elements to surface geometry and topology

Tyson Brochu, Christopher Batty, Robert Bridson
2010 ACM Transactions on Graphics  
Constructing the simulation mesh in this way allows us to place samples anywhere in the computational domain; we exploit this by choosing samples that accurately capture the geometry and topology of the  ...  Abstract We introduce an Eulerian liquid simulation framework based on the Voronoi diagram of a potentially unorganized collection of pressure samples.  ...  Acknowledgements This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.  ... 
doi:10.1145/1778765.1778784 fatcat:m4syph3kjbb4nndkrazwxrlram

How much Geometry it takes to Reconstruct a 2-Manifold in ℝ3 [chapter]

Daniel Dumitriu, Stefan Funke, Martin Kutz, Nikola Milosavljevic
2008 2008 Proceedings of the Tenth Workshop on Algorithm Engineering and Experiments (ALENEX)  
Facing the always present problem of round-off errors that easily compromise the exactness of those predicate decisions, an exact and robust implementation of these algorithms is far from being trivial  ...  and typically requires the employment of advanced datatypes for exact arithmetic as provided by libraries like CORE, LEDA or GMP.  ...  In step three, we essentially compute a discrete analogue of the geometric Voronoi diagram but using G(V ) as a discrete approximation of the space between the subsample vertices in S.  ... 
doi:10.1137/1.9781611972887.7 dblp:conf/alenex/DumitriuFKM08 fatcat:qwzqacziozcu3cm7chahqfpazy

How much geometry it takes to reconstruct a 2-manifold in R3

Daniel Dumitriu, Stefan Funke, Martin Kutz, Nikola Milosavljević
2009 ACM Journal of Experimental Algorithmics  
Facing the always present problem of round-off errors that easily compromise the exactness of those predicate decisions, an exact and robust implementation of these algorithms is far from being trivial  ...  and typically requires the employment of advanced datatypes for exact arithmetic as provided by libraries like CORE, LEDA or GMP.  ...  In step three, we essentially compute a discrete analogue of the geometric Voronoi diagram but using G(V ) as a discrete approximation of the space between the subsample vertices in S.  ... 
doi:10.1145/1498698.1537597 fatcat:4d2epmrzdzdepm3z4hchghaw3y

Curve-Skeleton Properties, Applications, and Algorithms

Nicu D. Cornea, Deborah Silver, Patrick Min
2007 IEEE Transactions on Visualization and Computer Graphics  
We also give a taxonomy of methods and analyze the advantages and drawbacks of each class of algorithms.  ...  In this paper, we provide an overview of many curve-skeleton applications and compile a set of desired properties of such representations.  ...  We would also like to thank the reviewers for their useful comments and suggestions. The source code and all models used in the experimental section can be downloaded from [139].  ... 
doi:10.1109/tvcg.2007.1002 pmid:17356219 fatcat:5fjr5p4jofcwplq2fksu5nb5z4

3D Voronoi Skeletons and Their Usage for the Characterization and Recognition of 3D Organ Shape

M. Näf, G. Székely, R. Kikinis, M.E. Shenton, O. Kübler
1997 Computer Vision and Image Understanding  
The proposed algorithm is a 3D generalization of the the surface of individual brains (i.e., the structure of sulci Voronoi skeleton concept, which is already in routine use for 2D and gyri) is fingerprint-like  ...  algorithm is given for the necessary regulariza-• Analysis and characterization of the morphological tion of the resulting Voronoi diagram.  ...  Diagram Generation 3D VORONOI SKELETONS Maximal sampling of the boundary of 3D objects on a 3.1.  ... 
doi:10.1006/cviu.1997.0610 fatcat:4i7sxdlrg5fa7i42fvkk4yct5u

MEASURING THE SHAPES OF MACROMOLECULES – AND WHY IT MATTERS

Jie Li, Paul Mach, Patrice Koehl
2013 Computational and Structural Biotechnology Journal  
Difference between sphere overlap and sphere contact. Let us consider for illustration four circles Si, Sj, Sk, and Sl in the plane.  ...  The edge cick (shown as a dashed line) is not part of this dual complex and therefore the circles Si and Sk are not in contact, according to our definition based on the alpha shape theory.  ...  Herbert Edelsbrunner, Institute of Science and Technology (1ST), Austria; we thank him for his rnenrorship and guidance. We thank Antoine Koehl for carefully reading the manuscript.  ... 
doi:10.5936/csbj.201309001 pmid:24688748 pmcid:PMC3962087 fatcat:52jd6kskm5ebdmdgsnnwzvtmqa

3D Skeletons: A State-of-the-Art Report

Andrea Tagliasacchi, Thomas Delame, Michela Spagnuolo, Nina Amenta, Alexandru Telea
2016 Computer graphics forum (Print)  
Informally, skeletons are descriptors which jointly describe the geometry, topology, and symmetry properties of a shape in compact and intuitive ways, providing a mean to capture the 'essence' of a shape  ...  Abstract Given a shape, a skeleton is a thin centered structure which jointly describes the topology and the geometry of the shape.  ...  Acknowledgments This work was partially funded by the advanced grant no. 291184 EXPRESSIVE from the European Research Council (ERC-2011-ADG_20110209), and by the H2020 EC Project GRAVITATE (RE-FLECTIVE7  ... 
doi:10.1111/cgf.12865 fatcat:dpyylsn5kffwxkjw4cza3zzcyu

Geometric Detection Algorithms for Cavities on Protein Surfaces in Molecular Graphics: A Survey

Tiago Simões, Daniel Lopes, Sérgio Dias, Francisco Fernandes, João Pereira, Joaquim Jorge, Chandrajit Bajaj, Abel Gomes
2017 Computer graphics forum (Print)  
Using the three-dimensional structure of a given protein (i.e., atom types and their locations in 3D) as retrieved from a PDB (Protein Data Bank) file, it is now computationally viable to determine a description  ...  This survey falls in the scope of molecular graphics and modeling, i.e., a research area at the intersection of computational biology, bioinformatics, computational geometry and computer graphics.  ...  Firstly, one constructs the Voronoi diagram of the atomic centers, as illustrated in Fig. 20(a) . • Computation of alpha spheres.  ... 
doi:10.1111/cgf.13158 pmid:29520122 pmcid:PMC5839519 fatcat:5t4rhndtunhvtmkm32niknst7i

Stability and Homotopy of a Subset of the Medial Axis [article]

F. Chazal, A. Lieutier
2004 Solid Modeling  
The Medial Axis has applications in image analysis and mathematical morphology, Solid Modeling, or domain decomposition for CAD to CAE (i.e. Finite Elements) models generation.  ...  Even if this is not explicit in other algorithms based on a 3D Voronoi diagram, topological fidelity in reconstruction algorithms is related to the ability to select a relevant subset of the Voronoi Diagram  ...  Proposed algorithms for the computation of the Medial axis in the first model of computation includes Voronoi Diagram of a finite set of rational points, and seams to be feasible for exact (that is rational  ... 
doi:10.2312/sm.20041396 fatcat:ghekirf3afcqvkgtclyixfpowi
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