SmartPhone Geometry. Jonathan Choate Groton School [email protected] www.zebragraph.com. WHAT’S COMING?. Why are they called Smart Cell phones? How do you figure out how far you are from home using your latitude and longitude?
Typical Cell Clusterhttp://www.google.com/patents?id=nO8tAAAAEBAJ&dq=martin+cooper
In order to serve the most customers the average cellhttp://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol1/pr4/article1.html#Cellssize is roughly 10 square miles. Each cell can service approximately 70-80 users at once because - Each cell is alloted 832 frequencies or channels - 42 channels are used for control issues- 790 are available for voice and data transmission.- Cell phones are duplex devises and need 2 frequencies per user unlike walkie talkies.
- Each cell is surrounded by 6 other cells so in order to avoid interference issues there has to be seven separate sets of frequencies.
- 395/7 is roughly 76 so for each cell there are 76 sets of frequencies so each cell can handle 76 users at once.
- In a 7 cell cluster, 532 people can be handled.
Cell Area =
2H avoid interference issues there has to be seven separate sets of frequencies.
Let H = distance between two centers of adjacent hexagons. avoid interference issues there has to be seven separate sets of frequencies.
Let Ru = Distance between two cells with same set of
frequences. Using the law of cosines, you get
C avoid interference issues there has to be seven separate sets of frequencies.
Let Rc be the cluster radius. Rc = AB =AD=DE and AE = Ru,
The area of the cluster can be calculated in two ways. Let C be the number of cells in the cluster
Therefore, C = 7
This shows that the possible cluster configurations contain i2 + j2 +ij cells where i and j are the displacements used to get to the nearest cell that can have the same set of frequencies.
i=3, j = 2 = 0
9 + 4 + 6 = 19
<PCQ = your longitude surface of the Earth?
<PCG = your latitude
R is the radius of the Earth
R = 3,959 Miles
Z= GP = R sin(lat)
X= PQ = PC sin(long)
Y = CQ=PC cos(long)
Activity 2. surface of the Earth? How do you calculate the distance between two points on the surface of the Earth given their latitude and longtitude? How far are you from home?
A possible arrangement of [email protected] towers??? surface of the Earth?
Tower maps can be found at http://www.towerco.com
Method 1: Trilaterization
To construct the surface of the Earth?Symmedian Point S for triangle ABC
2. Reflect G about the angle bisector of angle BAC creating point G1. Create ray AG1 and hide the angle bisector.
3. Repeat for vertices B and C, creating
rays BG2 and CG3.
4. Rays AG1, BG2 and CG3 are concurrent at the Symmedian point S
Part 4:Given Tower Locations How Can You Predict Coverage? surface of the Earth?
One solution to this problem is to construct for each tower the region formed by all the points nearest to that tower. These are called Vornoi Regions.
The 2 Tower Case tower the region formed by all the points nearest to that tower. These are called Vornoi Regions.
The 3 Tower Case tower the region formed by all the points nearest to that tower. These are called Vornoi Regions.
T tower the region formed by all the points nearest to that tower. These are called Vornoi Regions.1
Activity 4: Construct the Vornoi Regions for the five towers shown on the diagram below.
Given a line D, the directrix and a point F, the Focus, not on D, the set of points
equidistant from F and D form a parabola.
Here’s how to implement Fortune’s Algorithm using Geometer’s Sketchpad
Step 1. Open a file and select the graph option. Plot the points which represent the tower locations
Step 2. Construct a vertical line with a movable point D. Through D construct a
horizontal line. This will serve as a movable Directrix.
Step 3. Create the function which will plot the parabola with the upper most tower point and plot it.
References Geometer’s Sketchpad
Tracking GPS Sattelites
Cell and Cluster Design Information
Some Interesting Problems
Best GPS Information
Tower Location Maps