Voronoi diagrams – inventor, method, applications

Abstract

The article presents the person and works of Georgy Voronoi (1868-1908), the inventor of an original method of diagrams, a student of the famous mathematician Andrey Markov. Georgy Voronoi graduated from the Department of Physics and Mathematics at the University of St. Petersburg, and subsequently worked as a professor of mathematics at the Imperial University of Warsaw. One of his students was the future outstanding Polish mathematician Wacław Sierpiński. In his brief lifetime G. Voronoi published several important scientific articles on number theory. In an almost 100 page paper in French published in 1908 he described a method of diagrams, or polygons, which became known as the method of Voronoi diagrams. In the digital age this method and its modifications found new applications. The entry “Voronoi” is getting more popular on the Internet, and the method of Voronoi diagrams and its modifications are widely described in handbooks and scientific articles. The article presents application of the method in the most popular computer programs from the Geographic Information System (GIS) group and presents examples of its usage in research on geographic space in various scientific disciplines.

Multiplexed Imaging Analysis of the Tumor-Immune Microenvironment Reveals Predictors of Outcome in Triple-Negative Breast Cancer

“…Voronoi polygons have been used previously to define spatial organization and cellular morphology overall 28,36 . Each cell's Voronoi polygon is created from the location of its centroid; its polygon will border some number of polygons from other cells 37 . These borders can be used to model cell-to-cell interactions ( Figure 1d); cells whose polygons share a border can be considered adjacent (Figure 3a).…”

“…We used Voronoi diagrams to model cellular adjacencies within the TIME. Voronoi tessellation divides a planar space into a number of regions such that each point in the plane has its own region in the tessellation 37 . The sides of each Voronoi polygon are constructed to bisect two input points.…”

DigitalExposome: Quantifying the Urban Environment Influence on Wellbeing based on Real-Time Multi-Sensor Fusion and Deep Belief Network

“…The method of Voronoi visualisations is a computational geometry algorithm which allows the visualisation of large data sets [33]. The concept works by defining a set of polygon regions called cells, whereby the cells give an indication of the overall density of an object area of the size of the object itself [34].…”

“…Voronoi Diagram divides the space into a set of regions called Voronoi cells, including the space that is closest to the object (route location, in our case). The size of these cells gives an indication of the density of the area a certain object is in or the size of an object [34]. The cell structure also shows the Delaunay triangulation, which easily allows calculating an object's immediate set of neighbours.…”

Cartographic modelling of administrative divisions in the “Historical Atlas of Poland”

“…It is one of the methods of automatic area tessellation and change of the discrete-point model into a continuous-surface model (K.E. Brassel, D. Reif 1979;W. Pokojski, P. Pokojska 2018).…”