Real World Voronoi Diagrams

Autor:Joe Wilaj, Tim Brzezinski

In this activity you will learn about applications of Voronoi Diagrams and how to find Voronoi Lines. A Voronoi Diagram uses a set of points to divide up a 2D surface into regions called cells. Each cell corresponds to point in the set. A cell is the space on the surface that is closest to a given point.

Click the Voronoi Diagrams link above and read the applications section. Which application was suprising or interesting to you?

One example where this is useful is defining school districts. Plot a set on points on the map that represent the approximate location of each school. Then use the list tool to place the points in a list. Then type Voronoi(l1) in the input bar.

Does the Voronoi Diagram match closely with the actual districts? Where does it differ the most? Which district has the most neighbors?

The points in the set are called the seeds. Where should the seeds be placed in this diagram? On the location of the schools? the town centers? Move the seeds around to see how the regions change.

Voronoi Lines are Perpendicular Bisectors between two points. Construct the perpendicular bisector for seeds A and B in the window below.

Use the midpoint tool on AB. What do you notice?

Plot a point anywhere on the Perpendicular Bisector. Measure its distance from A and from B. Drag the point up and down the line. What do you notice?

Look at the drawing of perpendicular lines below. Drag the large points around. What do you notice about the relationship between the slopes?

Click Here to open a Graspable Math Activity to Learn how to draw Voronoi Lines on a graph

In your project, you will need cut off the lines where they intersect. For example to cut of the graph of y = 3x+1 between x = -4 and x = 0, use Function(y=3x+1, -4,0])

For your project, you will choose a real world situation to draw a Voronoi Diagram. Some Ideas: Plot the locations of all restaurant chains. Plot the locations of players on a field as in https://www.geogebra.org/m/kcmgrqn8v First you will paste a picture into a New GeoGebra Classic File. Then plot the Voronoi Seeds. Then you will work to come up with the equations of your Voronoi Lines as shown in this activity. Last you can use to code above to cut the lines and finish the diagram

Real World Voronoi Diagrams

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