Two views of the same point set
Give a solver a set of seed points, and there are two classic ways to turn them into a mesh. A Delaunay triangulation connects the seeds into triangles (tetrahedra in 3D) such that no seed lies inside any triangle’s circumscribing circle (or sphere). A Voronoi diagram does the opposite: it partitions space into one cell per seed, where each cell is the region closer to that seed than to any other.
The duality
The two are geometric duals of one another:
- Each Voronoi cell surrounds exactly one Delaunay seed.
- Each Voronoi vertex sits at the circumcenter of a Delaunay triangle.
- Each Voronoi face is the perpendicular bisector between two connected seeds.
So a Voronoi mesh is, in effect, the polyhedral dual of a Delaunay tet mesh — which is why Voronoi cells are naturally many-faced polyhedra.
Why Voronoi meshing is hard: clipping
Delaunay meshing is mature and widely used, but raw Voronoi cells have a problem at domain boundaries: a cell built from nearest-seed distance does not know where the surface is, so cells spill across walls and faults. The traditional fix is clipping — cutting each boundary cell against the geometry. Clipping is fragile: it creates sliver faces, very short edges, and degenerate cells right where quality matters most.
VoroCrust: Voronoi without clipping
The advance that made conforming Voronoi meshing practical is VoroCrust (Abdelkader, Ebeida, et al., ACM Transactions on Graphics, 2020). Instead of clipping, VoroCrust samples seed points in mirrored pairs across the surface so that Voronoi faces fall exactly on the boundary. It is described as the first provably correct algorithm for conforming polyhedral Voronoi meshing of non-convex, non-manifold domains, with quality guarantees on both surface and volume cells.
How AutoMesh-Geo helps
AutoMesh-Geo applies this clipping-free, VoroCrust-style approach so Voronoi cells conform to faults and features directly, rather than being cut to fit after the fact.