Mathematics
Voronoi Diagrams and Delaunay Triangulations for Spatial Analysis
Quick fact
The Delaunay triangulation is the exact dual of the Voronoi diagram: if you draw a line between every pair of points whose Voronoi cells share a boundary, you get a triangulation that maximizes the minimum angle of all triangles.
Why this is interesting
Have you ever noticed the crack patterns in dried mud or the spots on a giraffe? They follow a hidden geometric rule that also helps your phone find the nearest tower and scientists map disease hotspots.
Read the full explanation
Understanding Voronoi Diagrams and Delaunay Triangulations for Spatial Analysis
Imagine you have a set of points scattered on a map—like cities, cell towers, or trees. A Voronoi diagram assigns every location on the map to the closest point. This partitions the map into cells around each point. The boundary between two cells is the set of locations that are exactly equidistant to the two points. For two points, that's a straight line; for three, it's a point—the circumcenter of the triangle they form.
A deeper explanation
The Voronoi diagram and Delaunay triangulation are two sides of the same coin: they are dual structures. The Delaunay triangulation connects the original points so that each triangle's circumcircle contains no other points in its interior. This property ensures the triangulation avoids skinny, sliver triangles. The dual relationship means that for every edge in the Delaunay triangulation, there is a corresponding edge in the Voronoi diagram perpendicular to it, and vice versa. This duality allows us to compute one quickly from the other. The Voronoi diagram answers proximity questions—what's the nearest hospital?—while the Delaunay triangulation reveals the underlying connectivity of the points, useful for creating meshes, interpolating data, or simulating physical systems.