Voronoi Diagram Generator
Generate Voronoi diagrams from random or placed points, switch the distance metric, and run Lloyd relaxation to see irregular cells become evenly sized.
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One rule produces the pattern that appears everywhere
A Voronoi diagram assigns every location to its nearest seed point, and nothing more. The cell boundaries fall exactly halfway between neighbouring seeds, which is why they are always straight lines under ordinary distance. The same construction describes giraffe markings, soap foam, crystal grain boundaries, mobile phone coverage areas and which post office serves an address, because all of them are the answer to the same question about nearest points.
Changing the metric changes the geometry entirely
Under Manhattan distance the boundaries become staircases and cells acquire corners at forty-five degrees, because "equally far" means something different when you can only travel along axes. Under Chebyshev distance they turn into squares. The seeds do not move, and the picture is unrecognisable, which is a useful demonstration that the diagram is a property of the distance function rather than of the points.
Lloyd relaxation turns noise into an even texture
Repeatedly moving each seed to the centre of its own cell and recomputing makes the cells converge towards equal size and roughly hexagonal shape. A handful of iterations transforms a clumpy random scatter into something that looks deliberately arranged, which is why the technique is used for stippling, mesh generation and blue-noise sampling. The convergence is fast at first and then very slow, so three or four iterations gets most of the benefit.
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Frequently Asked Questions
What is a Voronoi diagram?
A partition of space where every location belongs to whichever seed point is nearest. The boundaries fall exactly halfway between neighbouring seeds.
Why are the edges straight?
Because the set of points equidistant from two seeds is the perpendicular bisector of the line between them, which is a straight line under ordinary distance.
What does Lloyd relaxation do?
It moves each seed to the centre of its own cell and repeats, which drives the cells towards equal size and roughly hexagonal shape. Most of the benefit arrives in three or four iterations.
Why do other metrics look so different?
Because the diagram is a property of the distance function rather than of the points. Manhattan distance produces staircase boundaries and Chebyshev produces squares, from the same seeds.
Where does this appear in practice?
Giraffe markings, soap foam, crystal grain boundaries, mobile coverage areas, nearest-facility assignment and blue-noise sampling. All are answers to the same nearest-point question.
Privacy & Security
Everything runs in your browser; nothing is uploaded.
How to Use
Set the point count and generate a diagram.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.