voronoi
Here are some utilities for creating and displaying Voronoi Cells.
Demo
Live demo utilizing R Shiny: https://olimay.shinyapps.io/MidtermVoronoi/
It displays the cell generation points in red and the cell boundaries in black.
Note: The source code for the demo above reuses code, with many thanks, developed by course staff for the Fall 2020 Harvard Extension School course, MATH E-151: Classic Mathematics with a Modern User Interface that is not part of this repository.
Voronoi Diagram
About Voronoi Diagrams
A Voronoi Diagram is a partitioning of a plane into separate cells containing a particular point called the "site" of the cell. The boundaries of each cell are constructed so that all points in a particular cell are closer to that cell's "site" as they are to any other cell's site. These cells are called Voronoi Cells [5].
The Voronoi Diagram is named after Russian mathematician Georgy Feodosevich Voronoi.
I am extremely new to this topic, having stumbled on the term while looking for information on regular plane tilings.
According to Wikipedia [2]:
They are used in many areas of science, such as the analysis of spatially distributed data, having become an important topic in geophysics, meteorology, condensed matter physics, and Lie groups. These tessellations are widely used in many areas of computer graphics, from architecture to film making and video games.
About this repository
These source files include functions for generating Voronoi Cells for a set of points, functions for generating a random set of points, and examples of tests.
The R code was originally developed as part of a midterm/final project for the Fall 2020 Harvard Extension School course, E-151, Classic Mathematics with a Modern User Interface.
It can generate two types of Voronoi Cells for two distance metrics: Euclidean distance (equivalently known as L2) and Manhattan distance (known as L1).
The algorithm used to generate the cells was adapted from this assignment from the University of Washington [1].
I adapted C++ code for computing spatial relationships of points and line segments from Dan Sunday's page on computational geometry [4].
I figured out how to generalize certain concepts used in the algorithm, particularly that of "perpendicular bisector", to Taxicab Geometry/Manhattan Distance with help from Aubrey Kemp's 2018 dissertation at Georgia State University [3].
References
- Anderson, Ruth. "Algorithm for generation of Voronoi diagrams." CSE326: Data Structures. https://courses.cs.washington.edu/courses/cse326/00wi/projects/voronoi.html (accessed Oct 11, 2020).
- "Georgy Voronoy." Wikipedia. https://en.wikipedia.org/wiki/Georgy_Voronoy (accessed Oct 14, 2020).
- Kemp, Aubrey, "Generalizing and Transferring Mathematical Definitions from Euclidean to Taxicab Geometry." Dissertation, Georgia State University, 2018. https://scholarworks.gsu.edu/math_diss/58
- Sunday, Dan. "Line and Plane Intersection (2D & 3D)." Geometry Algorithms. http://geomalgorithms.com/a05-_intersect-1.html (accessed Oct 14, 2020).
- "Voronoi Diagram." Wikipedia. https://en.wikipedia.org/wiki/Voronoi_diagram (accessed Oct 14, 2020).