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Divide each rectangle into regions so that each region represents the collection of all the points for which the site (A, B or C) is the closest of all sites.
Voronoi Diagrams:� Geometric Approach
Divide the waters around these three islands into Voronoi cells which fairly divide the waters by proximity to each island.
Voronoi Diagrams:� Geometric Approach
Divide the waters around these three islands into Voronoi cells which fairly divide the waters by proximity to each island.
Meteorologists in Boulder County, Colorado in the US track the cm of precipitation (rainfall and snowfall) received at three sites, marked on the map below:
1. Which station would best predict the precipitation of Lyons? Nederland? Jamestown? Create a Voronoi diagram to support your answers.
2. The meteorologists get additional funding to place a fourth precipitation collection site.
a. If this station is to be within the area between the other three but as far as possible from all three, then where should it be placed?��b. Where should they place this site if they want the most accurate data possible for the whole county?
Voronoi Diagrams:� Geometric Construction Approach� (see next slide)
Example 1
Example 2
Beekeepers can share the location of their apiaries (clusters of beehives) on the internet. This can be useful to prevent overcrowding, as each beehive needs sufficient territory to survive. Below is a map of part of Melbourne, Australia with apiaries marked in orange.
Which apiaries have the smallest territories? Which have the largest?
Example 3
A new post office delivery centre is to be opened in Warsaw, Poland, and the government is considering two locations for it (labelled A and B) in the diagram.
The government decides it would like to choose the location that will service a larger area.
Example 4
The Rega organization provides emergency helicopter services to the country of Switzerland, except for Valais Canton. It has 14 bases placed throughout the country, as shown in the map
When an emergency call is received, a base can get a helicopter flying within four minutes. Helicopters travel 400 km/hr on average. Rega’s objective “is to be able to reach any location in Switzerland–except in the canton of Valais–within 15 minutes’ flying time.” Can Rega reach its goal of responding to an accident anywhere in the country within 15 minutes?
How to�Create a �Voronoi�Diagram
Backwards:�
Based on �this Voronoi�Diagram, �where do�you think�the 3 seeds�are ?
Backwards: Where are the 4 seeds ?
New Seed?�
Where can�a new seed�be placed
that is �furthest�from the �current�4 seeds ?
Voronoi Diagrams:� Algebraic Approach
As an owner of three coffee shops in San Francisco, you want to open a fourth at a location that is as far as possible from the current three.
Voronoi Diagrams:� Algebraic Approach
a. State the midpoint of AB.
Current Locations:�A(12,6), B(12, 1.4), C(4,3)
b. State the equation of the perpendicular bisector of AB.
c. Find the midpoint of AC.
d. Find slope of AC.
e. Create an equation for the�perpendicular bisector of AC.
f. Repeat steps c. – e. in order to find the equation for the perpendicular bisector of BC.
h. Graph the lines to complete the Voronoi diagram and mark the proposed location, D.
g. Find the point that will be at the corner of all three cells (regions) of the Voronoi diagram.
Sample� Context: A food desert is an area that has limited access to affordable and nutritious food, in contrast with an area with higher access to supermarkets or vegetable shops with fresh foods, which is called a food oasis. Impoverished neighborhoods are predominantly located in food deserts.
Aim: Create a grocery store Voronoi diagram for the city of Nashville in order to expose the location of a prominent food desert in the city.
�Use the diagram to propose the location of a new grocery store.
Find the Practice Assignment on the Lesson Calendar.