Power diagrams: properties, algorithms and applications
SIAM Journal on Computing
On the computational geometry of pocket machining
On the computational geometry of pocket machining
Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
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
On the definition and the construction of pockets in macromolecules
Discrete Applied Mathematics - Special volume on computational molecular biology DAM-CMB series volume 2
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
Simulation Modeling and Analysis
Simulation Modeling and Analysis
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
Dynamic maintenance and visualization of molecular surfaces
Discrete Applied Mathematics - Special issue: Computational molecular biology series issue IV
Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d-dimensional space
Computer Aided Geometric Design
Proximity and applications in general metrics
Proximity and applications in general metrics
Efficient computation of continuous skeletons
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
Euclidean Voronoi diagram of 3D balls and its computation via tracing edges
Computer-Aided Design
Medial Axis Transformation of a Planar Shape
IEEE Transactions on Pattern Analysis and Machine Intelligence
Interaction interfaces in proteins via the Voronoi diagram of atoms
Computer-Aided Design
Multi-resolution protein model
ICCSA'07 Proceedings of the 2007 international conference on Computational science and Its applications - Volume Part II
Topologies of surfaces on molecules and their computation in O(n) time
Computer-Aided Design
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Voronoi diagrams have several important applications in science and engineering. While the properties and algorithms for the ordinary Voronoi diagrams of point sets have been well-known, their counterparts for a set of spheres have not been sufficiently studied. In this paper, we present properties and two algorithms for Voronoi diagrams of 3D spheres based on the Euclidean distance from the surface of spheres. Starting from a valid initial Voronoi vertex, the edge-tracing algorithm follows Voronoi edges until the construction is completed. The region-expansion algorithm constructs the desired diagram by successively expanding the Voronoi region of each sphere, one after another, via a series of topology operations, starting from the ordinary Voronoi diagram for the centres of spheres. In the worst-case, the edge-tracing algorithm takes O(mn) time, and the region-expansion algorithm takes O(n3 log n) time, where m and n are the numbers of edges and spheres, respectively. It should, however, be noted that the worst-case time complexity for both algorithms reduce to O(n2) for proteins since the number of immediate neighbor atoms for an atom is constant. Adapting appropriate filtering techniques to reduce search space, the expected time complexities can even reduce to linear. Then, we show how such a Voronoi diagram can be used for solving various important geometric problems in biological systems by illustrating two examples: the computation of surfaces defined on a protein, and the extraction and characterization of interaction interfaces between multiple proteins.