![]() ![]() This is illustrated in the figure below, in which I have plotted the first 30 iterations of the Lloyd’s algorithm for a random set of points. By moving the mouse over the image, one can explore how they are affected by a new point.įigure 12: Steps in Lloyd’s relaxation algorithmĪfter a few iterations, the cells will already have a “rounder” aspect and points will be more evenly distributed. In the Figure below, a Voronoi diagram (black) and Delaunay triangulation (grey) is plotted from a set of points. ![]() In mathematical terms, the Delaunay triangulation is the dual graph of the Voronoi diagram. If you take each of the points from a Voronoi diagram and link it with the points in its neighbouring cells, you will obtain a graph called Delaunay triangulation. Since the Voronoi diagrams partitions the space in polygons containing the closest points to each seed, the edges of Voronoi cells correspond exactly to the decision boundaries of a simple 1-nearest neighbour problem. The algorithm uses the \(k\) closest examples in the training dataset to make value predictions. That is to say, if you take any two points within a cell, the line that connects the two points will lie entirely within the cell.įinally, it should also be noted that Voronoi diagrams are tightly linked with the k-nearest neighbours algorithm (k-NN) - a very popular algorithm in classification, regression and clustering problems. It also has the advantage of generating Voronoi cells that are convex. Suppose \(P=\\]įigure 8: Comparison of Voronoi diagrams using the Euclidean (left) and Manhattan (right) distance for a same set of pointsĮuclidean distance is the most common distance measure in scientific applications of the Voronoi diagram. However, the same type of structure can be generalised to an \(n\)-dimensional space. So far, we have presented a simple two-dimensional Voronoi diagram. Mathematical definition and some interesting properties Interested reader may refer to this paper, in which the authors use Voronoi diagrams to model computer rendering of spots on animals coats. Over the course of the gestation these cells release melanin - hence spots radiate outward. Giraffe embryos have a scattered distribution of melanin-secreting cells, which is responsible for the dark pigmentation of the giraffe’s spots. For instance, this explains giraffe exhibit such patterns. Secondly, Voronoi diagrams are a spontaneous pattern whenever something is growing at a uniform growth rate from separate points (see Figure 2). This is very convenient if you are trying to squeeze as much as possible in a limited space - such as in muscle fibres or bee hives. As we mentioned earlier, Voronoi diagram completely tessellates the plan: hence, all space is used. These patterns are everywhere!Ī first reason for their omnipresence is that they form efficient shapes. From microscopic cells in onion skins, to the shell of jackfruits and the coat of giraffes. In Figure 3, I made a small collage of some naturally occurring Voronoi-like patterns. The pattern created by Voronoi diagrams is a common one in nature. Voronoi patterns are ubiquitous Voronoi patterns in nature In other words, all the area enclosed in the cell is closest to the point in the cell than to any other point.įigure 1: Voronoi diagram from 100 random points in a plane As you can see, every point is enclosed in a cell, whose boundaries are exactly equidistant between two or more points. As an illustration, in Figure 1, I plotted 100 random points and their corresponding Voronoi diagram. This produces a tessellation that completely covers the plane. Suppose you have \(n\) points scattered on a plane, the Voronoi diagram of those points subdivides the plane in exactly \(n\) cells enclosing the portion of the plane that is the closest to the each point. Voronoi diagram are simple, yet they have incredible properties which have found applications in fields ranging from cartography, biology, computer science, statistics, archaeology, all way to architecture and arts. You have encountered them thousands of times, but maybe did not call it this way. Voronoi diagrams (also known as Dirichlet tesselation or Thiessen polygons) are everywhere in nature. ![]()
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