Regions of nearest influence: A Voronoi diagram divides a plane into cells around a set of points called sites.
Every point inside a cell is closer to that cell''s site than to any other site.
| Term | Meaning |
|---|---|
| Site | One of the given points (e.g. hospital, shop) |
| Cell | Region of all points nearest to that site |
| Edge | Part of the perpendicular bisector between two neighbouring sites |
| Vertex | Point equidistant from 3 or more sites |
Interactive: see Voronoi cells, their bisector edges, and nearest-site highlighting.
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Construction steps: To construct a Voronoi diagram by hand: (1) For each pair of neighbouring sites, find their perpendicular bisector.
(2) The edges of the Voronoi diagram are segments of these bisectors.
(3) Each edge stops where it meets another bisector (at a Voronoi vertex).
Worked example — three sites
Sites are A(0, 0), B(6, 0), C(3, 4).
Describe the Voronoi diagram.
Step by step
- Find bisector of AB. Midpoint (3,0), gradient AB = 0 → bisector is vertical: x = 3.
- Find bisector of AC. Midpoint (1.5, 2), gradient AC = 4/3, perp grad = −3/4. Equation: y−2 = −(3/4)(x−1.5).
- Find bisector of BC. Midpoint (4.5, 2), gradient BC = −4/3, perp grad = 3/4. Equation: y−2 = (3/4)(x−4.5).
- The three bisectors meet at a single point (the circumcentre of triangle ABC).
Final answer
The Voronoi diagram has 3 cells, 3 edges, and 1 vertex (the circumcentre).
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Nearest site = cell membership: To find which cell contains a given point P, calculate its distance to each site and choose the smallest.
Alternatively, locate P on the Voronoi diagram.
Worked example — find nearest site
Sites are A(1, 1), B(7, 1), C(4, 6).
Point P = (4, 2).
Which site is nearest?
Step by step
- Distance PA.
- Distance PB.
- Distance PC.
- PA = PB < PC, so P lies on the boundary between cells A and B.
Final answer
P is equidistant from A and B — it lies on their shared Voronoi edge.
IB-style question — nearest-neighbour assignment
A weather app gives each town the reading of its NEAREST station. Stations: P(0, 0) = 18°, Q(6, 0) = 22°, R(3, 5) = 15°.
A town is at (4, 1). Which reading does it get?
Step by step
- Find the distance from the town to each station (no need to simplify — just compare).
- √5 is the smallest, so the town is in station Q's Voronoi cell.
Final answer
The town gets Q's reading, 22°.
Each Voronoi cell is the region closest to one site. A point takes the reading of whichever site's cell it falls in — here, the nearest station.
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Typical IB exam tasks: IB tasks include: (1) find which existing cell a new point belongs to, (2) find the Voronoi vertex, (3) identify which sites share a boundary, (4) find the perpendicular bisector equation for two given sites.
Worked example — Voronoi vertex
Sites A(0,0), B(4,0), C(2,4).
Find the Voronoi vertex (circumcentre of triangle ABC).
Step by step
- Bisector of AB: x = 2 (vertical).
- Bisector of AC: midpoint (1,2), grad AC = 2, perp grad = −1/2.
- At x = 2: y = −1 + 2.5 = 1.5.
Final answer
Voronoi vertex at (2, 1.5).
IB-style question — find a missing site from an edge
On a Voronoi diagram, the edge between sites P and Q is the line y = 4. Site P is at (1, 1).
Find the coordinates of site Q.
Step by step
- A Voronoi edge is the perpendicular bisector of the two sites, so the edge is the mirror line: P and Q are reflections of each other in y = 4.
- Reflecting in a horizontal line keeps x and flips the distance in y: the y-coordinate goes to 2(4) − 1.
Final answer
Q is at (1, 7).
A Voronoi edge is the perpendicular bisector of its two sites — so each site is the mirror image of the other across that edge.
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