This entry is part of the Chart Index, the reference library for the Chart Design Field Guide.

A Voronoi diagram partitions a plane into regions based on distance to a set of seed points. Every location in the plane belongs to the region of its nearest seed; the boundaries between regions are equidistant from neighbouring seeds. The form is beautiful — a quilt of polygons whose shapes are entirely determined by point positions — and surprisingly useful in visualisation, both as a structural form (showing service-area coverage, nearest-neighbour relationships) and as an interactive convenience (making sparse scatter plots easier to mouse over).

It is the chart most students encounter first as a fun geometric exercise and then never use professionally. In visualisation, it has three real applications: as a map of service coverage; as a layout algorithm for crowded points; and as an invisible hit-target overlay that makes interactive scatter plots more usable.

What it is

A Voronoi diagram of a set of points is a partition of the plane into convex polygons, one per seed point, where each polygon contains the set of locations closer to its seed than to any other seed. The boundaries are perpendicular bisectors of pairs of adjacent seeds. Variants include the weighted Voronoi (some seeds have larger regions) and the Voronoi treemap (Voronoi cells used in place of treemap rectangles).

The hero chart above plots twenty-four cafes on an inner-city grid, twelve kilometres by six, with each cafe's cell shaded by its weekly customers. The cells are the catchments: wherever you stand, the cafe whose cell you are in is the nearest one. The tight cluster of small cells in the middle is the city centre, where the busiest cafes compete for the same few blocks; the big pale cells on the edges are suburban locations that own a lot of ground and serve few people. Randomise the sites and the whole tessellation is recomputed — move one cafe and every neighbour's boundary shifts.

When to use it

Voronoi diagrams are the right choice when:

  • The data is point locations and the question is "what is the nearest-neighbour catchment of each?"
  • You are showing coverage or service area for facilities, towers, stations, sensors.
  • You need to make a sparse scatter plot interactive — Voronoi cells become hit targets.
  • You want a visually-striking spatial form for editorial display.
  • The points are distributed in a 2D plane with no overlap.

When not to use it

  • No spatial / locational meaning. Voronoi diagrams require positions in 2D space; abstract data with no spatial interpretation does not benefit.
  • Many points (1,000+). The diagram becomes a quilt of tiny cells; individual polygons are unreadable.
  • Hierarchical data. A treemap is the right form; Voronoi treemaps are aesthetic but lose the rectangular structure that makes treemaps comparable.
  • As a primary chart for executives. The form is unfamiliar to non-technical audiences; pair with a more conventional chart.

Design principles

Show the seed points

A Voronoi diagram without its seeds is geometrically complete but informationally orphaned. Always render the seeds — small dots — on top of the polygons so the reader can connect the partition to the data. The two figures below are the same cafes with and without them.

Cells without their sites: a quilt with no anchor
The hero's tessellation with the cafes removed. The cells are correct, but nothing tells the reader where the cafe inside each one is, so a cell cannot be read as a catchment — it is just a polygon.
Cells with their sites: every polygon has an owner
The same tessellation with the cafes drawn. Now each cell reads as the ground its cafe is nearest to, and the reader can see that a cafe's position inside its cell is rarely central — the boundaries are set by the neighbours, not by the site.

Use a quiet palette

Voronoi polygons are visually busy by nature. Leave the cells unfilled — as above — or fill them from a single hue with lightness driven by a domain attribute: catchment population, customer count, response time. The hero does the second with a seven-step teal ramp. A categorical rainbow across the cells turns the partition into a patchwork quilt in which the geometry is the only thing left to read.

Show the adjacency when neighbours matter

Two cells share an edge exactly when their seeds are nearest neighbours. Drawing those neighbour pairs as lines gives the Delaunay triangulation — the dual of the Voronoi diagram — and makes explicit which sites compete with which. Use it when the question is who borders whom rather than how much ground does each own.

Delaunay links: which cafes are neighbours
Each hairline joins two cafes whose cells share an edge. The dense web in the centre shows the city cafes competing on every side; the long spokes to the edge sites show that a suburban cafe has only three or four rivals, and they are far away.

Annotate the dominant cells

The largest polygons (or smallest, depending on the question) usually carry the story. Label them with their seed's name and a relevant metric — catchment area in square kilometres, customer count — and let the rest live in the tooltip.

Know when the cells become a texture

Past a few hundred seeds, the individual cells stop being readable and the diagram becomes a texture: useful as a hit-target layer, useless as a picture. At that density the honest chart is a heatmap or a hexbin.

Two hundred sites: the cells are a texture
Two hundred sensor stations across the same grid. The tessellation is still exact, but no cell can be identified, compared or labelled; all the reader can see is where the stations are dense and where they are sparse — which a dot map would show as well.

Pair with a relevant base map

For real-world spatial data, place the Voronoi cells over a faint base map showing roads, water, or boundaries. The partition gains context — why the cells have these shapes (rivers, mountain ranges, missing data regions).

Consider weighted Voronoi for unequal seeds

If the seeds have different weights (a flagship store vs. a small kiosk), the standard Voronoi gives them equal influence. Multiplicatively weighted Voronoi adjusts cell sizes by weight, producing a more realistic catchment model.

Document the distance metric

Standard Voronoi uses Euclidean distance — the straight-line distance. Real-world catchments may need Manhattan (grid-aligned) or travel-time distance. State the metric; the cells will look different.

Anatomy

The composition of a Voronoi diagram
An anatomical guide

A Voronoi diagram's anatomy is the seeds, their polygons, and the edges between them. The form's mathematical elegance carries the visual weight; the design choices are colour, label placement, and base-map context.

  • Delaunay triangulation — the dual graph of Voronoi; triangles connecting neighbouring seeds. Shown above as links.
  • Choropleth — region-based shading; different question (existing boundaries vs. computed catchments).
  • Hexbin / square grid — fixed cells regardless of point positions.
  • Treemap — the rectangular partition for hierarchical part-to-whole.
  • Scatter plot — the same points without the partition.
  • Buffer zones — fixed-radius regions around each point; the simpler alternative.

Reading list

  • Aurenhammer, F. (1991). Voronoi Diagrams — A Survey of a Fundamental Geometric Data Structure. The mathematical foundation.
  • Okabe, A., Boots, B., Sugihara, K. & Chiu, S. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. The application-focused reference.
  • Bostock, M., Ogievetsky, V. & Heer, J. (2011). D³: Data-Driven Documents. The library whose Voronoi overlay made the hit-target pattern standard.