AI HL · 12.8
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Voronoi diagrams in the real world
12.8.1
Drag the sites, add one or take one away, and watch the cells change. Drag the orange point P to see which site is nearest to it.
Drag a site or the point P.
Four clinics are at \(A(2, 3)\), \(B(9, 3)\), \(C(5, 9)\), \(D(11, 10)\), where the units are kilometres. The Voronoi diagram is drawn, with three homes P, Q and R. Each home goes to the nearest clinic. How many of the homes go to clinic B?
Have a go first. Then open the working one line at a time.
P is in the cell of A; Q and R are in the cell of B
A home in a cell is nearer to that cell's site than to any other site.
\(PA=\sqrt{5}=2.24\); \(PB=\sqrt{40}=6.32\)
P is closer to A, as the diagram shows.
2 homes go to B
Q and R are in the cell of B.
Three fire stations are at \(A(2, 3)\), \(B(8, 3)\), \(C(5, 11)\), where the units are kilometres. A house is at \(H(5, 5)\). The house is on an edge of the Voronoi diagram. Find the distance from H to station A.
Have a go first. Then open the working one line at a time.
\(HA=\sqrt{13}=3.61\)
Use the distance between two points.
\(HB=\sqrt{13}=3.61\)
The same, so H is on the edge between the cells of A and B.
\(HC=6\)
C is much further away. Only A and B matter here.
A and B are equally near
The house can use either station. The model cannot choose between them.
Three weather stations are at \(A(2, 2)\), \(B(9, 3)\), \(C(5, 9)\), where the units are kilometres. The rainfall last month was 14 mm at A, 22 mm at B and 17 mm at C. A farm at \(F(6, 5)\) uses the rainfall at its nearest station. Find the rainfall used for the farm.
Have a go first. Then open the working one line at a time.
\(FA=5\); \(FB=\sqrt{13}=3.61\); \(FC=\sqrt{17}=4.12\)
Work out the distance from the farm to every station.
B
B has the smallest distance. The farm is in the cell of B, as the diagram shows.
22 mm
Use the rainfall at B.
A town has 5 cafés. Describe how a Voronoi diagram models the situation. A new café then opens. How many cells does the new diagram have?
Have a go first. Then open the working one line at a time.
The 5 cafés
Each café is a site.
One cell for each café
A cell is every address that is closer to that café than to any other.
Which café is nearest to this address?
Find the cell that contains the address.
Distance is in a straight line
Roads, rivers and walking routes are ignored.
6 sites, so 6 cells
The new café takes a cell from the cells around it, so the old cells get smaller.