What is a weighted Voronoi diagram?
A standard Voronoi diagram assumes every site is equally attractive, so each border runs exactly halfway between two sites. Real places are rarely equal: a hospital with more beds, a radio mast with a stronger signal or a city with a larger population pulls from further away. A weighted Voronoi diagram gives each site a weight and measures distance in a way that lets heavier sites claim more space.
Three ways to weight a Voronoi diagram
- Multiplicatively weighted (Apollonius-style): weighted distance = distance ÷ weight. Borders become circular arcs, and a heavy site can surround a light one completely, leaving the light site an island.
- Additively weighted: weighted distance = distance − radius. Borders are branches of hyperbolas. Think of each site as a circle of the given radius; the cells are the regions closest to each circle's edge.
- Power diagram (Laguerre–Voronoi): weighted distance = distance² − radius². Borders stay straight lines, cells stay convex, and the diagram is fast to compute, so it is widely used in computational geometry and for weighted Voronoi treemaps.
With all weights equal, all three reduce to the ordinary Voronoi diagram. Turn on Overlay unweighted borders to compare.
How the cells are computed here
Curved and disconnected cells cannot be built with straight half-planes, so this tool assigns every pixel to the site with the smallest weighted distance. Fast resolution evaluates a coarser grid while you drag; Sharp gives smoother borders. For the classic straight-line diagram on a real map, use the Voronoi map generator; to change the distance rule itself instead of the weights, try Voronoi distance metrics.
Frequently asked questions
What is a weighted Voronoi diagram?
In a normal Voronoi diagram every site competes on equal terms, so borders lie halfway between sites. In a weighted Voronoi diagram each site has a weight that makes it reach further (or less far), so heavier sites claim more territory — like a big supermarket drawing customers from further away than a corner shop.
What is the difference between the three types?
Multiplicative weights divide the distance by the weight, which produces curved borders (circular arcs) and can even create cells with holes. Additive weights subtract the weight from the distance, giving hyperbolic borders. Power diagrams subtract the squared radius from the squared distance; their borders stay straight, which is why they are popular in computation.
Why does a site have no cell?
In additive and power diagrams a light site next to a much heavier one can be completely swallowed: every location is closer, in weighted terms, to the heavy site. The site still exists but owns no area.
How do I change a weight?
Click a site to select it, then use the weight slider. With a mouse you can also hover a site and scroll.