Topology Representation for the Voronoi Diagram of 3D Spheres

Euclidean Voronoi diagram of spheres in 3-dimensional space has not been explored as much as it deserves even though it has significant potential impacts on diverse applications in both science and engineering. In addition, studies on the data structure for its topology have not been reported yet. Presented in this paper is the topological representation for Euclidean Voronoi diagram of spheres which is a typical non-manifold model. The proposed representation is a variation of radial edge data structure capable of dealing with the topological characteristics of Euclidean Voronoi diagram of spheres distinguished from those of a general non-manifold model and Euclidean Voronoi diagram of points. Various topological queries for the spatial reasoning on the representation are also presented as a sequence of adjacency relationships among topological entities. The time and storage complexities of the proposed representation are analyzed.

[1]  Marina L. Gavrilova,et al.  Proximity and applications in general metrics , 1999 .

[2]  Deok-Soo Kim,et al.  Voronoi diagram of a circle set from Voronoi diagram of a point set: I. Topology , 2001, Computer Aided Geometric Design.

[3]  F M Richards,et al.  Areas, volumes, packing and protein structure. , 1977, Annual review of biophysics and bioengineering.

[4]  Y. Kawazoe,et al.  Computational Materials Science: From Ab Initio to Monte Carlo Methods , 2000 .

[5]  Deok-Soo Kim,et al.  Euclidean Voronoi diagram of 3D balls and its computation via tracing edges , 2005, Comput. Aided Des..

[6]  Erik Brisson,et al.  Representing geometric structures in d dimensions: topology and order , 1989, SCG '89.

[7]  Herbert Edelsbrunner,et al.  On the Definition and the Construction of Pockets in Macromolecules , 1998, Discret. Appl. Math..

[8]  Young Choi.  Vertex-based boundary representation of nonmanifold geometric models , 1989 .

[9]  A. Goede,et al.  Voronoi cell: New method for allocation of space among atoms: Elimination of avoidable errors in calculation of atomic volume and density , 1997 .

[10]  Jacques Chomilier,et al.  Nonatomic solvent‐driven voronoi tessellation of proteins: An open tool to analyze protein folds , 2002, Proteins.

[11]  Dan M. Bolser,et al.  Using convex hulls to extract interaction interfaces from known structures , 2004, Bioinform..

[12]  Jean-Daniel Boissonnat,et al.  Sur la complexité combinatoire des cellules des diagrammes de Voronoï Euclidiens et des enveloppes convexes de sphères de , 2022 .

[13]  José L. F. Abascal,et al.  The Voronoi polyhedra as tools for structure determination in simple disordered systems , 1993 .

[14]  Marina L. Gavrilova,et al.  Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d-dimensional space , 2003, Comput. Aided Geom. Des..

[15]  V. Luchnikov,et al.  Voronoi-Delaunay analysis of voids in systems of nonspherical particles. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  V. P. Voloshin,et al.  Void space analysis of the structure of liquids , 2002 .

[17]  M. L. Connolly.  Solvent-accessible surfaces of proteins and nucleic acids. , 1983, Science.

[18]  M. L. Connolly.  Analytical molecular surface calculation , 1983 .

[19]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[20]  David P. Dobkin,et al.  Primitives for the manipulation of three-dimensional subdivisions , 1987, SCG '87.

[21]  Kunwoo Lee,et al.  Principles of CAD/CAM/CAE Systems , 1999 .

[22]  Yasushi Yamaguchi,et al.  Nonmanifold topology based on coupling entities , 1995, IEEE Computer Graphics and Applications.

[23]  Valerio Pascucci,et al.  Dynamic maintenance and visualization of molecular surfaces , 2003, Discret. Appl. Math..

[24]  Donguk Kim,et al.  Voronoi diagram as an analysis tool for spatial properties for ceramics , 2002 .

[25]  Hans-Martin Will.  Practical and efficient computation of additively weighted Voronoi cells for applications in molecular biology , 1998 .

[26]  F. Richards.  The interpretation of protein structures: total volume, group volume distributions and packing density. , 1974, Journal of molecular biology.

[27]  Pascal Lienhardt,et al.  Subdivisions of n-dimensional spaces and n-dimensional generalized maps , 1989, SCG '89.

[28]  Kunwoo Lee,et al.  Partial Entity Structure: A Compact Boundary Representation for Non-Manifold Geometric Modeling , 2001, J. Comput. Inf. Sci. Eng..

[29]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.

[30]  C. Frömmel,et al.  The automatic search for ligand binding sites in proteins of known three-dimensional structure using only geometric criteria. , 1996, Journal of molecular biology.