Which cell does the new site fall in?: When a new site D is added: (1) Identify which existing cell contains D.
(2) Only that cell gets subdivided.
(3) Draw perpendicular bisectors between D and each neighbouring site in that cell.
Interactive: see how adding a site changes the diagram.
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Only adjacent cells change: Cells that were not neighbours of the old cell containing D are unaffected.
This is an important IB exam point.
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Worked example — adding site D
Existing sites: A(0,0), B(6,0), C(3,5).
A new site D(3,1) is added.
Which cells are affected and what new bisectors must be drawn?
Step by step
- Step 1: Identify which existing cell contains D(3,1). Calculate distances.
- DA = DB < DC, so D is on the boundary between cells A and B. Both cells A and B are affected.
- Step 2: Draw perpendicular bisector of AD. Midpoint (1.5, 0.5), gradient AD = 1/3, perp grad = −3.
- Step 3: Draw perpendicular bisector of BD. Midpoint (4.5, 0.5), gradient BD = −1/3, perp grad = 3.
Final answer
Cells A and B are subdivided. Two new bisectors replace the old A–B edge.
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Maximise distance from all sites: The point that is farthest from all sites (the safest location for toxic waste) is a Voronoi vertex — the centre of the largest empty circle.
Find all vertices, compute their distance to the nearest site, and choose the maximum.
Worked example — toxic waste location
Sites: A(0,0), B(8,0), C(4,6).
The Voronoi vertex is at V = (4, 1.67).
Find the radius of the largest empty circle.
Step by step
- Distance from V(4, 1.67) to A(0,0).
- By definition VA = VB = VC at a Voronoi vertex, so radius ≈ 4.33.
Final answer
The largest empty circle has centre V(4, 1.67) and radius ≈ 4.33 units.
| Context | Sites | Cell means |
|---|---|---|
| Hospitals | Hospital locations | Nearest hospital zone |
| Mobile towers | Tower locations | Area served by each tower |
| Supermarkets | Store locations | Catchment area for each store |
| Weather stations | Station locations | Area where that station''s reading applies |
IB exam framing: IB questions often ask: 'A new hospital is opened at D.
Draw the updated Voronoi diagram' or 'Find the optimal location for a new facility farthest from existing ones.' Always link back to perpendicular bisectors and equidistance.