from scipy.spatial import Voronoi, voronoi_plot_2d Since all of the cells are on the edge with no possible intersections, there are no finite polygons.īefore we add one more point to the square to get a finite region, let’s show the Python code to generate these simple diagrams. within a square), or it is equally distant to at least two corners (i.e. As for an interpretation, this diagram labels all of 2D space into two categories: any point on the plane is either closest to one of four corners in the unit square (i.e. Seeds are colored as blue dots at the corners of the unit square, dotted lines represent edges of an infinite polygon, and orange dots are Voronoi vertices. Each corner will be a seed, so there will be four Voronoi cells. To illustrate what Voronoi diagrams are and how to make them, let’s start with a very simple dataset: the four corners of the unit square. In practice, Voronoi cells are usually represented through their polygon vertices (Voronoi vertices) rather than by a collection of points this makes sense if you consider the inefficiency of directly storing points. a collection of points with a boundary, composed of points closest to the region’s seed. Regions were polygonal subsets of 2D space, i.e. While we won’t cover the algorithms to find the Voronoi polygon vertices, we will look at how to make and customize Voronoi diagrams by extending the functionality.īefore, Voronoi diagrams were defined as the concatenation of regions (Region Eq.) generated by the seeds. With an idea of what Voronoi diagrams are, we can now see how to make your own in Python. This extension occurs because regions are defined as all points nearest to a single seed and not the others so, since 2D space has infinite coordinates, there will always be infinite regions. Importantly, note that if the region’s generator is on the edge of the seed space and has no possible intersections, it will extend to infinity. The Voronoi Diagram is the concatenation of n regions generated by the seeds. It’s important to recall that a generator, D k, comes from the input data, whereas points in its region, R k, is the output. In English, the equation is “This region is equal to the set of points in 2D space such that the distance between any one of these points and this generator is less than the distance between the point and all other generators.” To fully understand the math, be sure to map the words to every symbol in the equation. For each seed k in D, a region R k is defined by the Region equation. Given a distance metric dist and a dataset D of n 2-dimensional generator coordinates, a Voronoi diagram partitions the plane into n distinct regions. Cell boundaries shared between two regions signify the space that is equally distant, or “equidistant”, to the seeds. A Voronoi cell, or Voronoi “region” denotes all of the space in the plane that is closest to its seed. Each seed generates its own polygon called a Voronoi “cell,” and all of 2-dimensional (2D) space is associated with only one cell. The set of points that generate the Voronoi diagram are called “seeds” or “generators” for they generate the polygon shapes in practice, the seeds are your cleaned data. The diagram’s structure is a data-driven tessellation of a plane and may be colored by random or to add additional information. Voronoi Diagrams are an essential visualization to have in your toolbox.
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