Visualizing Spatial Data: Delaunay Triangulations and Voronoi Diagrams with the deldir Package in R

elaunay Triangulations / Voronoi Diagram

The deldir package in R is a powerful tool for performing Delaunay triangulations and generating Voronoi diagrams. It is useful for spatial analysis and geometric modeling. In this article, I will provide an overview of its functionalities, how to use them, and an explanation of the concepts of Delaunay triangulation and Voronoi diagrams.

Fundamental Concepts

Voronoi Diagram:
A Voronoi diagram divides space into regions, where each region corresponds to a point in a set of points. All points within a region are closer to the corresponding point than to any other point.

Delaunay Triangulation:
Delaunay triangulation is a triangulation of a set of points that maximizes the minimum angle of all the triangle's angles, avoiding "thin" triangles. Each triangle has a circumscribed circle that does not contain other points in the set.

Using the deldir Package

Installation and Loading

First, install and load the deldir package:

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install.packages("deldir")
library(deldir)
  

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Main Functions

deldir

The primary function to perform Delaunay triangulation and create a Voronoi diagram:

voronoi_data <- deldir(x, y)
  

Where x and y are vectors containing the coordinates of the points.

tile.list

Extract Voronoi cells from the deldir object:

tiles <- tile.list(voronoi_data)
  

plot.deldir

Plot the Voronoi diagram and/or the Delaunay triangulation:

plot(voronoi_data)
  

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Converting to a SpatialLinesDataFrame

Converting to SpatialLinesDataFrame:

• The tile.list function is used to convert the voronoi_data object into a list of polygons, where each polygon represents a Voronoi cell.
• A voronoi_lines list is created to store the polygon lines.

Creating the SpatialLinesDataFrame Object:

• A loop iterates over each polygon in the voronoi_tiles list.
• For each polygon, the coordinates are extracted and added to the voronoi_lines list as a Lines object.
• All lines are then converted into a SpatialLines object and finally into a SpatialLinesDataFrame, with line IDs assigned automatically.

Creating the Leaflet Mapa

library(leaflet)

map <- leaflet() %>%
 addTiles() %>%
 addProviderTiles(providers$Esri.WorldImagery) %>%
 addCircles(lng = ignition_points$LONGITUDE, lat = ignition_points$LATITUDE, radius = 500, color = "red", fill = TRUE, fillOpacity = 0) %>%
 addPolylines(data = voronoi_spdf, color = "yellow", weight = 1)

map
  

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Explanation

• Converting to SpatialLinesDataFrame:
  • tile.list converts the voronoi_data object into a list of polygons, with each polygon representing a Voronoi cell.
  • A loop iterates over each polygon in the list, extracting the coordinates and adding them to the voronoi_lines list as a Lines object.
  • These lines are then converted into a SpatialLines object and finally into a SpatialLinesDataFrame, with IDs assigned automatically.
• Creating the Leaflet Map:
  • A leaflet map is created using addTiles() to add a default set of base tiles.
  • Satellite imagery tiles from Esri are added with addProviderTiles(providers$Esri.WorldImagery).
  • Ignition points are added to the map as red circles using their longitude and latitude coordinates.
  • Voronoi diagram lines are added as yellow polylines.

Displaying the Map

The map is displayed using the map object, providing a clear visualization of the Voronoi diagram overlaid on a satellite map.