Lloyd Relaxation of Voronoi Diagrams
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
A Voronoi diagram is a tessellation in which each polygon (or "Voronoi cell") represents the set of points closest to a central "seed" point. Voronoi diagrams have a variety of uses in multiple fields including physics, chemistry, mathematics, map generation, and climatology. Lloyd relaxation (also called Voronoi relaxation) redraws these polygons after moving the seed of each cell in a Voronoi diagram to the centroid of that cell. Repeated iteration of this process causes the seeds to space out evenly and settle into a stable state known as a "centroidal Voronoi tessellation". This Demonstration shows the effect of repeated Lloyd relaxation on a randomly generated Voronoi diagram. To step through the iteration, click "append relaxation" several times. Use the "diagram index" control to review the history of the iteration. To start again from scratch, reset with some chosen number of points.
Contributed by: Daniel Jacobson (January 2012)
Open content licensed under CC BY-NC-SA
"Lloyd Relaxation of Voronoi Diagrams"
Wolfram Demonstrations Project
Published: January 31 2012