SmartPhone Geometry
description
Transcript of SmartPhone Geometry
![Page 1: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/1.jpg)
SmartPhone Geometry
Jonathan Choate
Groton School
www.zebragraph.com
![Page 2: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/2.jpg)
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?
![Page 3: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/3.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
Part 1. Why are they called Cell Phones?
![Page 4: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/4.jpg)
http://www.google.com/patents?id=nO8tAAAAEBAJ&dq=martin+cooper
QuickTime™ and a decompressor
are needed to see this picture.
![Page 5: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/5.jpg)
![Page 6: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/6.jpg)
Typical Cell Cluster
![Page 7: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/7.jpg)
• http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol1/pr4/article1.html#Cells
![Page 8: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/8.jpg)
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.
![Page 9: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/9.jpg)
- 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.
![Page 10: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/10.jpg)
Activity 1. Given that cells are hexagonal in shape, what are the possible cluster sizes that insure no interference occurs between adjacent cells?
QuickTime™ and a decompressor
are needed to see this picture.
![Page 11: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/11.jpg)
Hexagon Geometry
R
32
R
Cell Area = 23 3
R2
![Page 12: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/12.jpg)
2H
1H
120 degreesQuickTime™ and a
decompressorare needed to see this picture.
Ru
![Page 13: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/13.jpg)
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
![Page 14: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/14.jpg)
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
![Page 15: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/15.jpg)
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
![Page 16: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/16.jpg)
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
![Page 17: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/17.jpg)
I = 1, j =1 I = 2, j = 0 I = 2 , j =1 I = 3 , j = 0
![Page 18: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/18.jpg)
i=3, j = 2
9 + 4 + 6 = 19
![Page 19: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/19.jpg)
Part 2. How do you assign coordinates to locations on the surface of the Earth?
![Page 20: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/20.jpg)
<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)
![Page 21: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/21.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
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?
![Page 22: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/22.jpg)
MA2029
Part 3: How does your phone figure out where you are?
![Page 23: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/23.jpg)
![Page 24: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/24.jpg)
![Page 25: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/25.jpg)
![Page 26: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/26.jpg)
![Page 27: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/27.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 28: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/28.jpg)
![Page 29: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/29.jpg)
A possible arrangement of AT@T towers??? Tower maps can be found at http://www.towerco.com
![Page 30: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/30.jpg)
• Activity 3 How does Assisted GPS determine your position? Method 1: Trilaterization
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
![Page 31: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/31.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 32: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/32.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 33: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/33.jpg)
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
![Page 34: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/34.jpg)
Method 2: A Slick Construction
QuickTime™ and a decompressor
are needed to see this picture.
![Page 35: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/35.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 36: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/36.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 37: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/37.jpg)
How about an algorithm for a general solution?
QuickTime™ and a decompressor
are needed to see this picture.
![Page 38: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/38.jpg)
Part 4:Given Tower Locations How Can You Predict Coverage?
![Page 39: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/39.jpg)
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.
![Page 40: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/40.jpg)
The 2 Tower Case
![Page 41: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/41.jpg)
The 3 Tower Case
![Page 42: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/42.jpg)
Activity 4: Construct the Vornoi Regions for the five towers shown on the diagram below.
T2
T1
T3
T4
T5
![Page 43: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/43.jpg)
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.
![Page 44: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/44.jpg)
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 )
![Page 45: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/45.jpg)
QuickTime™ and a decompressor
are needed to see this picture.QuickTime™ and a
decompressorare needed to see this picture.
![Page 46: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/46.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
The function you need to plot the parabola with focus at ( xF, yF ) and directrix the line y = yD
QuickTime™ and a decompressor
are needed to see this picture.
![Page 47: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/47.jpg)
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.
![Page 48: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/48.jpg)
![Page 49: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/49.jpg)
![Page 50: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/50.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 51: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/51.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 52: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/52.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 53: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/53.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 54: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/54.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 55: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/55.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
![Page 56: SmartPhone Geometry](https://reader034.fdocuments.net/reader034/viewer/2022051402/56815940550346895dc67ff6/html5/thumbnails/56.jpg)
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