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## PowerPoint Slideshow about ' SmartPhone Geometry' - ramiro

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

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

Let H = distance between two centers of adjacent hexagons.

Let Ru = Distance between two cells with same set of

frequences. Using the law of cosines, you get

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.

9 + 4 + 6 = 19

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

Tower maps can be found at http://www.towerco.com

Activity 3 How does Assisted GPS determine your position?

Method 1: Trilaterization

To construct the 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

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.

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.

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.

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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