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Voronoi Diagrams

A variety of questions about Voronoi Diagrams presented in a self-marking, online exercise.
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This is level 2: questions involving the vocabulary and mathematics of voronoi diagrams. You can earn a trophy if you get at least 7 questions correct and you do this activity online.

Voronoi Diagram

This Voronoi diagram is the floor plan of a theme park and the lettered sites are the restaurants.

1. How many cells does the diagram contain?

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2. How many vertices are around the cell containing site H ?

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3. How many edges does the diagram contain (do not include the edges on the perimeter of the rectangle)?

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4. The units on the axes represent kilometers. Calculate the distance between the restaurant at J(1.5,1) and the restaurant at F(5.5,1). Give your answer in kilometers.

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5. Calculate the distance between the restaurant at A(0.5,4.5) and the restaurant at E(4.5,1.5). Give your answe in kilometers.

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6. Find the coordinates of the point equidistant from the restaurants at E, C and D.

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7. What is the equation of the line equidistant from sites B and C.

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8. Each restaurant has equipment to measure the air quality index (AQI). The following table shows the current readings:
Air Quality Index
Use 'nearest neighbour interpolation' to determine the AQI at (1.5,2.5).

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9. Estimate the AQI at the point (4.5,0.25)

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10. The Voronoi diagram below shows four villages (marked as black dots). What are the coordinates of the best location for a waste dump if it has to be within the convex hull created by the four villages but as far away as possible from the villages (Largest Empty Circle).
Toxic Waste Dump Site

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This is Voronoi Diagrams level 2. You can also try:
Level 1 Level 3 Level 4

Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.
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Description of Levels

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Perpendicular Bisector - A step-by-step guide showing how to construct the perpendicular bisector of a line segment. This skill can then be used to construct a Voronoi Diagram.

Printable Worksheet: Get a real 'feel' for Voronoi diagrams with this resource on which you can draw Voronoi diagrams using the good, old-fashioned compasses, pencil and ruler method.

Level 1 - General questions about a diagram of a rectangular shaped island.

Level 2 - Questions involving the vocabulary and mathematics of Voronoi diagrams.

Level 3 - Find the coordinates of the missing sites in ten Voronoi Diagrams.

Level 4 - Define a region on a Voronoi Diagrams by finding the equations of the edges.

Exam Style Questions - A collection of problems in the style of IB exam paper questions.

More on Loci including lesson Starters, visual aids, investigations and self-marking exercises.

Voronoi move euclidean by Jahobr [CC0]

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

The video above is from Revision Village.

Definitions

Sites: These are the important locations from which the positions of the lines of the Voronoi diagram are calculated.

Cells: These are the areas that surround the sites and contain the points which are closer to that site than to any other site. The cells are labelled according to the site which they contain. They are also known as regions.

Edges: These are the borders between the cells. They are the lines showing the points equidistant from pairs of sites.

Vertices: A vertex is the point at which three or more edges meet. Each vertex is equally close to the sites whose cells meet at that vertex. They can also be called intersections.

Nearest Neighbour interpolation is a simple method of estimating the value of a variable at any point by using the variable's value at the nearest site.

Toxic Dump Problem: This problem can be described as finding the optimal position for a toxic waste dump, so as to maximise its distance from the nearest town. If not on the border of the diagram the location will always be at one of the vertices.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

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