Three recycling depots sit at A(0, 0), B(8, 0) and C(4, 6) on a map of a district, in kilometres. A waste transfer station has to go as far as possible from the nearest depot. Almost everyone puts it in the middle. The middle is 4.00 km from its nearest depot. The right answer is 4.33 km.
Each boundary is the perpendicular bisector of two sites. All three meet at one point, and that point is the vertex.
| Word | What it is |
|---|---|
| Site | One of the given points. The depots. In an exam the coordinates are given to you. |
| Cell | Everywhere that is nearer to one particular site than to any other. |
| Edge | A boundary between two cells. It is the perpendicular bisector of those two sites, because that is where the nearest one changes. |
| Vertex | Where three edges meet. It is the same distance from three sites at once. |
Everything in this sub-topic follows from one line of 3.5: a point on the perpendicular bisector of two sites is equally far from both. So a cell boundary is a bisector, and a point where three boundaries cross is equally far from three sites.
The centroid of the three depots is (4, 2). Measure from it:
The station is only as safe as its nearest depot, so the number that matters is 4.00. The 4.47s are wasted: it is too close to C and not close enough to anything else to make up for it.
The Voronoi vertex is at (4, 5/3), which is (4, 1.67). Measure again:
All three equal, because that is what a vertex is, and the nearest is 4.33. Better than 4.00 by a third of a kilometre. Moving away from C by a third of a unit buys more than it costs, and the balance point where it stops being worth it is exactly where all three distances are equal.
This is why the exam answer is always a vertex. Away from a vertex, one site is nearest and you can always do better by walking directly away from it. You can only stop when no single site is nearest, which means at least three are tied, which means you are standing on a vertex. The syllabus says so directly: in examinations the solution point is at an intersection of three edges.
For a "where do we put the dump" question:
Check the region. A vertex outside the district, or out at sea, is not an answer, and questions are often built so the obvious vertex is ruled out.
If each site carries a measurement, every point in its cell is estimated to have the same value. A rain gauge at A reading 92 mm means the whole of A's cell is estimated at 92 mm.
For the point (1, 1): to A is √2 = 1.41, to B is 7.07, to C is 5.83. A is nearest, so (1, 1) takes A's reading.
It is a crude estimate and saying so is often worth a mark: the value jumps at every boundary, and a point just inside A's cell gets A's number even if it is almost equally close to B.
A new depot opens at D(4, 1). Draw the bisectors between D and each nearby site, and rub out the parts of the old edges that are now inside D's cell.
The old vertex (4, 1.67) is only 0.67 from D and 4.33 from the others, so it is deep inside D's new cell. Adding a site can only take territory away, never give it: every other cell either shrinks or stays the same, and no new region appears anywhere.
The practical consequence is that a transfer station sited at the old vertex is now 0.67 km from a depot, which is a real planning point and the sort of thing the last part of a question asks you to comment on.
You are not asked to construct a Voronoi diagram from nothing. The syllabus is explicit: coordinates are given, and constructing perpendicular bisectors by hand is not required. What is required is finding a boundary's equation, saying which site is closest to a given point, working out a cell's area, and the dump problem.
A vertex is where two bisectors cross, so it is a pair of simultaneous equations, and that is the one thing here the machine does much faster than you.
When you may use it. Applications. A calculator is allowed in every paper.
The mark people lose. Answering with the centroid. It is a real point with a real name and it is 4.00 from its nearest depot where the vertex is 4.33, so it loses the accuracy mark and usually the reasoning mark with it. The other one is giving the vertex and stopping: the question asks how far the station is from the nearest depot, so quote the 4.33 as well, and say which three sites it is equidistant from.
1. For sites A(0, 0), B(8, 0) and C(4, 6), find the y-coordinate of the Voronoi vertex, to 2 decimal places.
2. How far is that vertex from the nearest site, to 2 decimal places?
3. How far is the centroid (4, 2) from its nearest site, to 2 decimal places?
4. A rain gauge at each site reads A 92 mm, B 78 mm, C 86 mm. Using nearest neighbour interpolation, what is the estimate at (1, 1), in mm?
5. A new site opens at D(4, 1). What happens to the three original cells?
1 markTwo correct bisector equations.
1 markSolving them to get the vertex.
1 markThe distance from the vertex to a site.
1 markSaying why it is the vertex: it is equidistant from three sites, and anywhere else you can move further from the nearest one.
That last mark is for the reasoning, and it is the one most often left blank. One sentence earns it.
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