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Standard level

13.02 Voronoi diagrams

Worksheet
Voronoi diagrams
1

Consider the following Voronoi diagram for sites \text{A, B} and \text{C}:

a

What can be said about all the points in cell \text{A}?

b

What can be said about all the points along the edge between cell \text{A} and cell \text{B}?

c

What can be said about the vertex?

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2

Consider the following Voronoi diagram:

a

Which site is the point (-3, 5) closest to? Explain your answer.

b

Verify your answer to part (a) by finding the distance between point (-3, 5) and each of the sites.

c

Identify the site which is closest to the following points:

i
(1, 4)
ii
(-3,0)
iii
(3,-3)
iv
(1,2)
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3

Consider the following Voronoi diagram:

a

Show that (-3,0) is equidistant to sites \text{A} and \text{B}.

b

Show that (3,-1) is equidistant to sites \text{B} and \text{D}.

c

Show that (5,-3) is equidistant to sites \text{B}, \text{C} and \text{D}.

d

Show that (-3,-3) is equidistant to sites \text{A}, \text{B} and \text{C}.

e

Does there exist a point that is equidistant to all four sites? Explain your answer.

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4

The following Voronoi diagram shows the locations of four schools in a city. Families are encouraged to send their children to their local (nearest) school. Each unit on the graph represents 1 \text{ km}.

a

Yosef's house is located at (3,0). Which school should he attend?

b

Ashleigh lives at (1, 4).

i

Which school should she attend?

ii

Ashleigh really wants to attend School \text{C} because that is where most of her friends are going. How much further will she have to travel to attend school \text{C}, compared to her local school? Round your answer to two decimal places.

c

Han lives at (4, 2).

i

Which school should he attend?

ii

Han and his family move 3 \text{ km} West. Which school should he attend now?

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5

The following Voronoi diagram shows a map of towns, where each unit is a kilometre. We need to find a location for a waste dump, however none of the towns want to be near it.

Suggest a location for the waste dump somewhere on this map. Explain your choice.

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