Voronoi diagrams, quasi-triangulations, and beta-complexes for disks in R2: the theory and implementation in BetaConcept

Abstract

Voronoi diagrams are powerful for solving spatial problems among particles and have been used in many disciplines of science and engineering. In particular, the Voronoi diagram of three-dimensional spheres, also called the additively-weighted Voronoi diagram, has proven its powerful capabilities for solving the spatial reasoning problems for the arrangement of atoms in both molecular biology and material sciences. In order to solve application problems, the dual structure, called the quasi-triangulation, and its derivative structure, called the beta-complex, are frequently used with the Voronoi diagram itself. However, the Voronoi diagram, the quasi-triangulation, and the beta-complexes are sometimes regarded as somewhat difficult for ordinary users to understand. This paper presents the twodimensional counterparts of their definitions and introduce the BetaConcept program which implements the theory so that users can easily learn the powerful concept and capabilities of these constructs in a plane. The BetaConcept program was implemented in the standard C++ language with MFC and OpenGL and freely available at Voronoi Diagram Research Center (http://voronoi.hanyang.ac.kr).

Structural characterization of the micropatterned egg plastron in the mosquito, Aedes albopictus

“…The BetaConcept program is a software designed to understand the concept of geometrical shapes, including Voronoi tessellation, quasi‐triangulation, and others (Kim et al . ). Using the BetaConcept program, we confirmed that most of the chorionic sculptures of A. albopictus eggshells have a typical “hexagon” partitioning surface pattern in accordance with Voronoi tessellation in a two‐dimensional plane.…”

BetaSCPWeb: side-chain prediction for protein structures using Voronoi diagrams and geometry prioritization

“… Voronoi diagram of 2D circular atoms and its derivative constructs (Figures drawn using BetaConcept ( 27 )). ( A ) The Voronoi diagram (blue lines) of circles and its quasi-triangulation (red lines), ( B ) the β-shape (shaded polygon plus line segments) and corresponding offset (red curve) for a probe of radius β 1 ≥ 0, ( C ) the corresponding β-complex (shaded triangles plus line segments) for β 1 , ( D ) the β-shape (shaded polygon) and the offset (red curve) for β 2 > β 1 , ( E ) the β-complex (shaded triangles) for β 2 and ( F ) the zero β-complex (shaded triangles plus line segments) for a β-probe of zero radius.…”

“…Figure 1F is called the zero beta-complex corresponding to the probe of zero radius and therefore it contains the intersection information among atoms. Figures 1A - 1F are drawn using BetaConcept ( 27 ). We emphasize that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathcal {BC}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathcal {BS}$\end{document} for an arbitrary β-value can be computed very efficiently through a binary search of the sorted simplex sets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathcal {QT}$\end{document} ( 23 ).…”

BetaCavityWeb: a webserver for molecular voids and channels

“…Figure 1(f) shows an example of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathcal {BC}$\end{document} for β -value. Note that Figure 1 was drawn by BetaConcept ( 38 ).…”