smartphone geometry jonathan choate groton school jchoate@groton.org

SmartPhone Geometry

Jonathan Choate

Groton School

jchoate@groton.org

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?

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

• Given tower locations how can you predict the coverage?

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Part 1. Why are they called Cell Phones?

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

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Typical Cell Cluster

• http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol1/pr4/article1.html#Cells

In order to serve the most customers the average cellsize 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.

Activity 1. Given that cells are hexagonal in shape, what are the possible cluster sizes that insure no interference occurs between adjacent cells?

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Hexagon Geometry

R

32

R

Cell Area = 23 3

R2

2H

1H

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Ru

Let H = distance between two centers of adjacent hexagons.

H = 3R

Let Ru = Distance between two cells with same set of frequences. Using the law of cosines, you get

Ru2 =(2H)2 +(1H)2 −2(2)(1)H2cos(120)

Ru= (2H)2 +(1H)2 +(2)(1)H2

Ru = 7HRu = 7 3R

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

<ADE=120

B

A

D

C

ERc

Rc

RcRc

Rc

Ru

Rc =Ru3

Ru= 3Rc

Rc =Ru3

Ru= 3Rc

The area of the cluster can be calculated in two ways. Let C be the number of cells in the cluster

2 23 3 3 3R 7R2 2

C =

Therefore, C = 7

23 3 R2

Areaof cluster =C

Areaof cluster =  3 32

Rc2

Rc =Ru3

=7 3R

3== 7R

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.

Ru2 = (iH)2+(jH)2-2(i)(j)H2cos(120)

Ru= (i)2+(j)2+(i)(j) H

Ru= (i)2+(j)2+(i)(j)  3R

I = 1, j =1 I = 2, j = 0 I = 2 , j =1 I = 3 , j = 0

i=3, j = 2

9 + 4 + 6 = 19

Part 2. How do you assign coordinates to locations on the surface of the Earth?

<PCQ = your longitude<PCG = your latitudeR is the radius of the EarthR = 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)

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Activity 2. 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?

MA2029

Part 3: How does your phone figure out where you are?

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A possible arrangement of AT@T towers??? Tower maps can be found at http://www.towerco.com

• Activity 3 How does Assisted GPS determine your position? Method 1: Trilaterization

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To construct the Symmedian Point S for triangle ABC

1. Construct the centroid G2. 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, creatingrays BG2 and CG3.4. Rays AG1, BG2 and CG3 are concurrent at

the Symmedian point S

Method 2: A Slick Construction

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How about an algorithm for a general solution?

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Part 4:Given Tower Locations How Can You Predict Coverage?

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

The 3 Tower Case

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

T2

T1

T3

T4

T5

Fortune’s Method

• 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.

y =yD

( x, y )F( xF , yF  )

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The function you need to plot the parabola with focus at ( xF, yF ) and directrix the line y = yD

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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.

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

Tracking GPS Satteliteshttp://www.n2yo.com/?s=36397

Cell and Cluster Design Informationhttp://www.wirelesscommunication.nl/reference/chaptr04/cellplan/reuse.htm

Some Interesting Problemswww.ece.ucdavis.edu/~chuah/classes/eec173B/eec173b-s09/.../hw1.pdf -

Fortune's Algorithmhttp://en.wikipedia.org/wiki/File:Fortunes-algorithm.gif http://www.ams.org/featurecolumn/archive/voronoi.html

Best GPS Informationhttp://www.u-blox.com/images/stories/Resources/gps_compendiumgps-x-02007.pdf

Tower Location Mapshttp://towerco.com

References