THE CATCHMENT AREA FOR YOUR BUSINESS

In a medical emergency, you always want to rush to the nearest hospital. A city has a number of hospitals. Suppose we have a map showing each hospital catchment area (for anyone in that region, that hospital is closer than any other) that will be highly helpful. How do you make that map?

It can be easily done. But doing it manually is difficult. With a computer algorithm, we can easily create a map.

Start with two hospitals at points A and B on the city map. Draw a line that connects them. Find the midpoint of that line and then draw a line that passes through that midpoint and is perpendicular to the line from A to B. That line divides the city into two regions. One region having A contains all the points closer to A than to B. The other contains all the points closer to B than to A.

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Now, look at the third hospital, at point C. Repeat the above process to the hospitals at A and C, you get a second line of separation. Repeat for B and C, you get the third line of separation. Erase the unnecessary extensions of the lines, you get the following figure.

We will continue to do for all the hospitals in the city. We get the picture given below. This kind of division of the map into regions is called a "Voronoi" diagram. It is named after the Russian mathematician Gregory Voronoi (1868-1908)

We will see one striking use of this diagram. In the 1950s, a Cholera outbreak occurred in London, killing 10% of the population. One physician by name John snow thought that cholera came from contaminated water supplies. He convinced others of his theory by first making the number of deaths at each address on the map. He then also identified the "catchment area" of a particular water pump at 40 Broadwick street. Points within the catchment area were closer to the broad street pump than any other pump. It turned out that almost all the deaths marked on the map lay inside the catchment area of the Broad Street pump.

This demonstrated clearly that contaminated water was indeed the cause of cholera. Today the spot where the pump once stood is marked by a memorial and there is a pub right next to it named in John snow's honor.

There are two ways to create a Voronoi diagram. We can either use crow flying distance (straight distance) to mark catchment areas or walking distance along streets and alleys (Manhatten distance) to carve out catchment areas. John snow used 'Manhatten distance".

You can make a Voronoi diagram for your business and similar businesses in the city. It will be helpful in designing a better marketing strategy. Wikipedia lists so many uses for Voronoi diagrams.

Brilliant mathematical ideas are a joy forever.

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