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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?

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Smartphone geometry
SmartPhone Geometry

Jonathan Choate

Groton School

[email protected]

www.zebragraph.com


What s coming
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?

  • How does your phone figure out where you are if there is no GPS reception?

  • Given tower locations how can you predict the coverage?



Http www google com patents id no8taaaaebaj dq martin cooper
http://www.google.com/patents?id=nO8tAAAAEBAJ&dq=martin+cooperhttp://www.google.com/patents?id=nO8tAAAAEBAJ&dq=martin+cooper


Typical cell cluster

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.


Hexagon geometry
Hexagon Geometry avoid interference issues there has to be seven separate sets of frequencies.

R

Cell Area =


2H avoid interference issues there has to be seven separate sets of frequencies.

1H

Ru

120 degrees


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.

Rc

B

D

Rc

Rc

Rc

Rc

E

Ru

A

Let Rc be the cluster radius. Rc = AB =AD=DE and AE = Ru,

<ADE=120


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

<GPR=<PQC=90

Z= GP = R sin(lat)

PC=R cos(lat)

X= PQ = PC sin(long)

=Rcos(lat)sin(long)

Y = CQ=PC cos(long)

= Rcos(lat)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?




  • Activity 3 surface of the Earth? How does Assisted GPS determine your position?

    Method 1: Trilaterization


To construct the surface of the Earth?Symmedian Point S for triangle ABC

  • Construct the centroid G

    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


Method 2 a slick construction
Method 2: A Slick Construction 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

T2

T3

T4

T5

Activity 4: Construct the Vornoi Regions for the five towers shown on the diagram below.


Fortune s method
Fortune’s Method tower the region formed by all the points nearest to that tower. These are called Vornoi Regions.

  • For more than 3 points, finding Vornoi Regions is very hard. In 1986, Steven Fortune came up with an ingenious way of finding them making use of parabolas.


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

Cell triangulation

http://searchengineland.com/cell-phone-triangulation-accuracy-is-all-over-the-map-14790

Tracking GPS Sattelites

http://www.n2yo.com/?s=36397

Cell and Cluster Design Information

http://www.wirelesscommunication.nl/reference/chaptr04/cellplan/reuse.htm

Some Interesting Problems

www.ece.ucdavis.edu/~chuah/classes/eec173B/eec173b-s09/.../hw1.pdf -

Fortune's Algorithm

http://en.wikipedia.org/wiki/File:Fortunes-algorithm.gif

http://www.ams.org/featurecolumn/archive/voronoi.html

Best GPS Information

http://www.u-blox.com/images/stories/Resources/gps_compendiumgps-x-02007.pdf

Tower Location Maps

http://towerco.com


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