Skip to main content

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.


 ADVT: TODAY'S DEALS IN AMAZON

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.  

Comments

Post a Comment

Popular posts from this blog

LISSAJOUS FIGURES

  Definition:  "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures".      We know sine wave motion and circular motion is basically same.  Hence we draw two circles A and B perpendicular to each other.  The circle B rotates twice faster than circle A.  That is, frequency of circle B is two times than that of A.        A particle at the intersection of two circles is subjected to two sine wave motion   A and B at 90 degree simultaneously.  The particle will describe figures depending on the frequency and phase of A and B .  In our case, the ratio of frequency is  1:2 and the two waves are in phase.        To draw lissajous figures :  A moving point in both the circles are chosen.   Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two.  Hence the points are marked on the circles according to their speed.  Then straight lines
Post a Comment
Read more

THE PARABOLA

          A jet of water shooting from a hose pipe will follow a parabolic path.  What is the so special about parabola.    Y= x^2 Draw a graph for the above equation.  It will result in a parabola.  This parabola is also called unit parabola.  Any equation involving square will yield a parabola. Example:  Y = 2x^2 +3x+3 (also called quadratic equation)    X= 2 and -2, both  satisfies the equation 4 = X^2.  Parabolic equations always have two solutions.     Any motion taking place freely under gravity follows parabolic path. Examples:   An object dropped from a moving train,   A bomb dropped from flying plane,  A ball kicked upwards.      If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point.  This fact is made use of in Dish Antenna, Telescope mirrors, etc.      Inverted parabola shape is used in the construction of buildings and bridges.  Because the shape is able to bear more weight.      A plane
Post a Comment
Read more

CASINO'S GAME

           Let us find out how the casino survives with mathematics.      Say, your friend invite you for a game of dice.  You must bet (wager) 2 dollars.  If you roll 'six' you will get back 8 dollars.  The game will go on for 30 rounds.  All sounds good.      The probability of rolling 'six' is 1/6.  Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5.  That is, you are expected to win 5 rounds out of 30.  Hence your gain will be 5 * 8 =40 dollars.  ok.  This also implies that you will loose 25 rounds.  Hence your loss will be 25*2 =50 dollars.  Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars.  There will be a loss of -1/3 or 0.33 dollars per round.  It is not a fair game.     Let us make a simple formula to calculate  'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect
4 comments
Read more