Quasi-worlds and quasi-operators on quasi-triangulations

Kim, Deok-Soo; Cho, Youngsong; Sugihara, Kōkichi

doi:10.1016/j.cad.2010.06.002

Cited by 51 publications

(27 citation statements)

References 22 publications

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“…A quasi-triangulation may have a hierarchy of worlds where there can be one or more small world(s) underneath the root world. [27,58,59] Being the subset of a quasi-triangulation, the zero beta-complex has a root world which may or may not contain one or more small world(s). By definition, the vdW-volume corresponding to the atoms in small worlds in the quasi-triangulation is a subset of the vdWvolume corresponding to the atoms of the root world.…”

Section: Interior Tetrahedron (Tetra4): β-Cellmentioning

confidence: 99%

Beta‐decomposition for the volume and area of the union of three‐dimensional balls and their offsets

et al. 2012

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Given a set of spherical balls, called atoms, in three-dimensional space, its mass properties such as the volume and the boundary area of the union of the atoms are important for many disciplines, particularly for computational chemistry/biology and structural molecular biology. Despite many previous studies, this seemingly easy problem of computing mass properties has not been well-solved. If the mass properties of the union of the offset of the atoms are to be computed as well, the problem gets even harder. In this article, we propose algorithms that compute the mass properties of both the union of atoms and their offsets both correctly and efficiently. The proposed algorithms employ an approach, called the Beta-decomposition, based on the recent theory of the beta-complex. Given the beta-complex of an atom set, these algorithms decompose the target mass property into a set of primitives using the simplexes of the beta-complex. Then, the molecular mass property is computed by appropriately summing up the mass property corresponding to each simplex. The time complexity of the proposed algorithm is O(m) in the worst case where m is the number of simplexes in the beta-complex that can be efficiently computed from the Voronoi diagram of the atoms. It is known in ℝ(3) that m = O(n) on average for biomolecules and m = O(n(2)) in the worst case for general spheres where n is the number of atoms. The theory is first introduced in ℝ(2) and extended to ℝ(3). The proposed algorithms were implemented into the software BetaMass and thoroughly tested using molecular structures available in the Protein Data Bank. BetaMass is freely available at the Voronoi Diagram Research Center web site.

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Section: Interior Tetrahedron (Tetra4): β-Cellmentioning

confidence: 99%

Beta‐decomposition for the volume and area of the union of three‐dimensional balls and their offsets

et al. 2012

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“…The Voronoi diagram VD of three-dimensional spheres can be computed by the edge-tracing algorithm taking O(n 3 ) time in the worst case but O(n) time on average for molecules [3]. The quasi-triangulation QT is obtained by transforming VD in O(n) time in the worst case [15], [7]. Then, the beta-complex BC is extracted from QT using a binary search in O(n log n + k) time in the worst case where k is the number of simplexes in the resulting beta-complex [6].…”

Section: Voronoi Diagrams and Their Derivative Constructs In Bull!mentioning

confidence: 99%

“…The Voronoi diagram of spheres is the generalization of the power diagram [14] which is a generalization of the ordinary Voronoi diagram of points from the L 2 -norm point of view. By the same token in the dual space, the quasi-triangulation is the generalization of the regular triangulation and the Delaunay triangulation [15], [7]. The beta-complex is similarly the generalization of the (weighted) alpha-shape [6], [16].…”

Section: Voronoi Diagrams and Their Derivative Constructs In Bull!mentioning

confidence: 99%

“…In this regard, there are rich prior studies on the geometry of molecules but mostly based on ad hoc approaches depending on discipline or even depending on a particular aspect of problem: No unified framework of theory on the geometry issue of molecular structure is currently available. Authors' group at the Voronoi Diagram Research Center (VDRC) [1], Hanyang University, has been developing the Molecular Geometry (MG) theory during the past decade based on the Voronoi diagram, particularly with the Voronoi diagram of spheres, and its derivative structures [2], [3], [4], [5], [6], [7], [8].…”

Section: Introductionmentioning

confidence: 99%

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Molecular Geometry and BULL!

Cho

Kim

Ryu

et al. 2014

2014 International Conference on Cyberworlds

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Geometric properties are critical for the function of molecules consisting of atoms which are usually modeled as a set of spheres in 3D. We propose "Molecular Geometry" which is a theoretical framework of computational understanding of the geometry of molecules in the claim that most, hopefully all, molecular structure problems can be effectively and efficiently facilitated by the "geometrization" of the problem at hand. We also report BULL!, the molecular geometry engine based on the Voronoi diagram of spheres, the quasi-triangulation, and the beta-complex. Being a program implemented in C++, application programmers can simply call API-functions of BULL! to create application programs correctly, efficiently, and conveniently. The BULL! engine, which will be freely available from the Voronoi Diagram Research Center at Hanyang University, is designed so that application programs are completely independent of future modifications and improvements.

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“…a vertex array, a tetrahedron array, and a gate array) (21) . For the definition and properties of the quasi-triangulation, see (21), (22).…”

Section: Quasi-triangulationmentioning

confidence: 99%

BetaMol: A Molecular Modeling, Analysis and Visualization Software Based on the Beta-Complex and the Quasi-Triangulation

Cho

Kim

Ryu

et al. 2012

JAMDSM

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Molecular shape is one of the most critical factors that determines molecular function. Therefore, it is frequently desirable to understand geometric characteristics of a molecule more precisely and efficiently. In this paper, we introduce the BetaMol, a molecular modeling, analysis, and visualization software based on the recent theory of the beta-complex and the quasi-triangulation that are derived from the Voronoi diagram of three-dimensional spherical atoms. The powerful features of the BetaMol are solely based on a unified, single framework of the mathematically rigorous and computationally efficient beta-complex theory. The BetaMol is implemented in the standard C++ language with OpenGL graphics library and freely available at Voronoi Diagram Research Center web site (http://voronoi.hanyang.ac.kr).

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Quasi-worlds and quasi-operators on quasi-triangulations

Cited by 51 publications

References 22 publications

Beta‐decomposition for the volume and area of the union of three‐dimensional balls and their offsets

Beta‐decomposition for the volume and area of the union of three‐dimensional balls and their offsets

Molecular Geometry and BULL!

BetaMol: A Molecular Modeling, Analysis and Visualization Software Based on the Beta-Complex and the Quasi-Triangulation

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