Voronoi diagrams in the real world Problem set B

Three libraries are at \(D(9, 9)\), \(E(1, 3)\), \(F(5, 1)\), where the units are kilometres. A student lives at \(P(4, 6)\). The student goes to the nearest library.

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 covered by weather stations X and Y. A point Z on the edge between the two cells is 9 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 city planner has a map of the city's supermarkets and its roads. Which of these can a Voronoi diagram answer? (i) The area in which each supermarket is the nearest one. (ii) The shortest driving route to a supermarket. (iii) Which supermarket is nearest to a given address, in a straight line. (iv) The average shopping bill at each supermarket.

Four bike-hire stands are at \(A(5, 2)\), \(B(1, 11)\), \(C(11, 3)\), \(D(1, 6)\), where the units are hundreds of metres. The Voronoi diagram is drawn, with three offices P, Q and R marked. A new stand opens at \(N(10, 2)\).

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

A district has several primary schools.

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 radio transmitters are at \(X(2, 9)\) and \(Y(10, 5)\), where the units are kilometres. A farm is at \(H(7, 9)\). A mountain lies between H and transmitter 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 transmitter for the farm, and suggest one way to improve the model.