Voronoi diagrams in the real world Problem set A

Three clinics are at \(A(2, 3)\), \(B(10, 2)\), \(C(6, 10)\), where the units are kilometres. A patient lives at \(P(5, 5)\). The patient goes to the nearest clinic.

1. (a) Work out the distance from P to each site.
2. (b) Which site's cell is P in?

A Voronoi diagram shows the areas served by fire stations X and Y. A point Z on the edge between the two cells is 7 km from station X. A point W lies in the cell of X.

1. (a) How far is Z from station Y?
2. (b) What can you say about the distance from W to station X compared with the distance from W to station Y?

A town council has a map of its schools, bus routes and rivers. Which of these can a Voronoi diagram answer? (i) The area closest to each school. (ii) The quickest route by bus between two schools. (iii) Which school is nearest to a given house, in a straight line. (iv) The number of pupils at each school.

Four cafés are at \(A(10, 11)\), \(B(10, 2)\), \(C(1, 3)\), \(D(4, 8)\), where the units are hundreds of metres. The Voronoi diagram is drawn, with three flats P, Q and R marked. A new café opens at \(N(2, 2)\).

1. (a) From the diagram, write down the nearest site to each of the flats P, Q and R.
2. (b) A new café opens at N. Work out which of the flats is nearer to N than to its old nearest site.

A town has several dental surgeries.

1. (a) Say what the sites are, what a cell shows, and one question the diagram answers.
2. (b) Give one thing the model leaves out.

Two phone masts are at \(X(2, 3)\) and \(Y(10, 3)\), where the units are kilometres. A house is at \(H(6, 7)\). A hill lies between H and mast X.

1. (a) Show that H is on the edge between the cells of X and Y.
2. (b) Explain why the Voronoi model may not give the best mast for the house, and suggest one way to improve the model.