Why is the empty circumcircle property important?

A triangulation is Delaunay iff every triangle’s circumcircle contains no other site. That criterion maximizes the minimum angle among triangulations of the point set, which helps avoid skinny triangles in meshes.

How are Voronoi and Delaunay dual?

Each Delaunay edge corresponds to two neighboring Voronoi cells; Voronoi vertices sit at circumcenters of Delaunay triangles. Drawing one determines the combinatorial structure of the other.

What does Bowyer–Watson do when I add a point?

It finds triangles whose circumcircles contain the new site, removes them, and retriangulates the resulting polygonal hole to the new point — restoring the Delaunay property incrementally.

Delaunay & Voronoi

Bowyer–Watson triangulation and dual Voronoi tessellation; click to add seeds, drag to move.

About this model

Given seed points in the plane, the Voronoi tessellation assigns each location to the nearest seed, while the Delaunay triangulation connects seeds whose Voronoi cells share an edge — equivalently, triangles whose circumcircles are empty of other seeds. This simulator builds the triangulation with a Bowyer–Watson incremental algorithm and draws the dual Voronoi diagram. Click to add seeds and drag to move them; the mesh and cells update live. The model is planar Euclidean nearest-neighbor geometry without weighted Voronoi sites, constrained triangulations, or curved metrics. Add and drag points to see empty-circumcircle flips and cell boundaries reshape.

Who it's for: Computational geometry, mesh generation, GIS, and intermediate algorithms courses.

Key terms

Delaunay triangulation
Voronoi diagram
Bowyer–Watson
Circumcircle
Tessellation
Nearest neighbor

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

Frequently asked questions

Why is the empty circumcircle property important?: A triangulation is Delaunay iff every triangle’s circumcircle contains no other site. That criterion maximizes the minimum angle among triangulations of the point set, which helps avoid skinny triangles in meshes.
How are Voronoi and Delaunay dual?: Each Delaunay edge corresponds to two neighboring Voronoi cells; Voronoi vertices sit at circumcenters of Delaunay triangles. Drawing one determines the combinatorial structure of the other.
What does Bowyer–Watson do when I add a point?: It finds triangles whose circumcircles contain the new site, removes them, and retriangulates the resulting polygonal hole to the new point — restoring the Delaunay property incrementally.

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