Power diagrams: properties, algorithms and applications
SIAM Journal on Computing
Three-dimensional alpha shapes
ACM Transactions on Graphics (TOG)
Spheres, molecules, and hidden surface removal
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Algorithmic geometry
Voronoi diagram of a circle set from Voronoi diagram of a point set: topology
Computer Aided Geometric Design
Voronoi diagram of a circle set from Voronoi diagram of a point set: geometry
Computer Aided Geometric Design
Computational Line Geometry
Computing Smooth Molecular Surfaces
IEEE Computer Graphics and Applications
On the combinatorial complexity of euclidean Voronoi cells and convex hulls of d-dimensional spheres
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Defining, Computing, and Visualizing Molecular Interfaces
VIS '95 Proceedings of the 6th conference on Visualization '95
Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d-dimensional space
Computer Aided Geometric Design
Weighted alpha shapes
Proximity and applications in general metrics
Proximity and applications in general metrics
An efficient algorithm for three-dimensional β-complex and β-shape via a quasi-triangulation
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Molecular surfaces on proteins via beta shapes
Computer-Aided Design
Triangulation of molecular surfaces
Computer-Aided Design
Region-expansion for the Voronoi diagram of 3D spheres
Computer-Aided Design
Computer-Aided Design
Euclidean Voronoi diagram of 3D balls and its computation via tracing edges
Computer-Aided Design
Manifoldization of β-shapes in O(n) time
Computer-Aided Design
Quasi-worlds and quasi-operators on quasi-triangulations
Computer-Aided Design
A β-shape from the Voronoi diagram of atoms for protein structure analysis
ICCSA'06 Proceedings of the 6th international conference on Computational Science and Its Applications - Volume Part I
On the shape of a set of points in the plane
IEEE Transactions on Information Theory
Topologies of surfaces on molecules and their computation in O(n) time
Computer-Aided Design
Querying simplexes in quasi-triangulation
Computer-Aided Design
QTF: Quasi-triangulation file format
Computer-Aided Design
Technical Section: Efficient construction of the Čech complex
Computers and Graphics
Geometry guided crystal phase transition pathway search
Computer-Aided Design
Anomalies in quasi-triangulations and beta-complexes of spherical atoms in molecules
Computer-Aided Design
Triangulating molecular surfaces on multiple GPUs
Proceedings of the 20th European MPI Users' Group Meeting
Protein structure optimization by side-chain positioning via beta-complex
Journal of Global Optimization
| Hi-index | 0.00 |
The proximity and topology among particles are often the most important factor for understanding the spatial structure of particles. Reasoning the morphological structure of molecules and reconstructing a surface from a point set are examples where proximity among particles is important. Traditionally, the Voronoi diagram of points, the power diagram, the Delaunay triangulation, and the regular triangulation, etc. have been used for understanding proximity among particles. In this paper, we present the theory of the @b-shape and the @b-complex and the corresponding algorithms for reasoning proximity among a set of spherical particles, both using the quasi-triangulation which is the dual of the Voronoi diagram of spheres. Given the Voronoi diagram of spheres, we first transform the Voronoi diagram to the quasi-triangulation. Then, we compute some intervals called @b-intervals for the singular, regular, and interior states of each simplex in the quasi-triangulation. From the sorted set of simplexes, the @b-shape and the @b-complex corresponding to a particular value of @b can be found efficiently. Given the Voronoi diagram of spheres, the quasi-triangulation can be obtained in O(m) time in the worst case, where m represents the number of simplexes in the quasi-triangulation. Then, the @b-intervals for all simplexes in the quasi-triangulation can also be computed in O(m) time in the worst case. After sorting the simplexes using the low bound values of the @b-intervals of each simplex in O(mlogm) time, the @b-shape and the @b-complex can be computed in O(logm+k) time in the worst case by a binary search followed by a sequential search in the neighborhood, where k represents the number of simplexes in the @b-shape or the @b-complex. The presented theory of the @b-shape and the @b-complex will be equally useful for diverse areas such as structural biology, computer graphics, geometric modelling, computational geometry, CAD, physics, and chemistry, where the core hurdle lies in determining the proximity among spherical particles.