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Home/Math Visualization/Delaunay & Voronoi

Delaunay & Voronoi

This interactive simulator explores Delaunay & Voronoi in Math Visualization. Bowyer–Watson triangulation and dual Voronoi tessellation; click to add seeds, drag to move. Use the controls to change the scenario; watch the visualization and any graphs or readouts to connect the model with lectures, labs, and homework.

Who it's for: Best once you already know the basic definitions and want to build intuition. Typical context: Math Visualization.

Key terms

  • delaunay
  • voronoi
  • delaunay voronoi
  • math
  • visualization

Display

Mode

20

Shortcuts

  • •Click empty space to add a seed
  • •Drag a yellow seed to move it

Measured values

seeds (sites)0
Delaunay triangles0
Delaunay edges0
Voronoi edges (interior)0

How it works

The Voronoi diagram of N seeds partitions the plane into cells of points closer to seed i than to any other. Its dual graph is the Delaunay triangulation: connect two seeds whenever their Voronoi cells share an edge. Delaunay maximises the minimum angle of all triangles (no other triangulation does as well) and has the empty-circumcircle property — toggle the option to verify that no other seed lies inside any circumcircle. Implemented with the Bowyer–Watson incremental algorithm.

Key equations

V(sᵢ) = { x : ‖x − sᵢ‖ ≤ ‖x − sⱼ‖ ∀ j }
Delaunay edge ⇔ V(sᵢ) and V(sⱼ) share an edge
empty-circle: no seed is strictly inside circumcircle of any triangle