A Voronoi diagram partitions a plane into regions according to which fixed point is nearest. Each region belongs to one site, while its boundaries consist of points that are equally close to two or more sites.
The Reasoning and Key Terms
Suppose several fire stations are represented by points. A Voronoi diagram divides the surrounding area so that every location is assigned to its nearest station.
| Term | Precise meaning |
|---|---|
| Site | A fixed point that generates the diagram, such as the position of a fire station. |
| Cell | The region containing every point closer to its site than to any other site. Outer cells may be unbounded. |
| Edge | A boundary between two cells. Every point on an edge is equidistant from the two corresponding sites, so the edge lies on their perpendicular bisector. |
| Vertex | A point where at least three edges meet. It is equidistant from three or more sites. |
For example, let two sites be and . Their midpoint is , so the perpendicular bisector is . Points to the left of this edge belong to the cell of , while points to the right belong to the cell of .
If a third site is added, its perpendicular bisectors with and meet the first edge at . This is a vertex because its distance from each site is .
A common misconception is that an edge joins two sites. This is incorrect: an edge is part of the perpendicular bisector between them.
Exam Technique
This is SL 3.6: Voronoi diagrams, studied by both AI SL and HL students and examinable in Paper 1 or Paper 2. An examiner may ask you to identify a cell, determine the nearest site, find an edge equation, use nearest-neighbour interpolation, or locate the largest empty circle at a vertex. Always justify answers using distance, cell membership, or perpendicular bisectors rather than visual appearance alone.