Global Positioning
A Lesson in the “Math + Fun!” Series
Jan. 2006
Global Positioning
Slide 1
About This Presentation
This presentation is part of the “Math + Fun!” series devised
by Behrooz Parhami, Professor of Computer Engineering at
University of California, Santa Barbara. It was first prepared
for special lessons in mathematics at Goleta Family School
during three school years (2003-06). “Math + Fun!” material
can be used freely in teaching and other educational settings.
Unauthorized uses are strictly prohibited. © Behrooz Parhami
Jan. 2006
Edition
Released
First
Jan. 2006
Revised
Global Positioning
Revised
Slide 2
We Need to Know Where We Are
A ship at sea may be lost when
we can’t determine its location
Jan. 2006
A plane flying over cloud-covered
mountains needs location info
Global Positioning
Slide 3
GPS Provides Location Information
For navigation:
Hiking,
Driving,
Flying,
Sailing
Jan. 2006
For assistance:
Rescue,
The visually
impaired
For tracking:
Earth,
Structures,
Children,
Autos,
Pets,
Wildlife
Global Positioning
For recreation:
Gaming,
Dining,
Sports
Slide 4
Locating a Point in
One Dimension
Hello, 911? I have had an
accident on Highway 101,
seven miles from Goleta.
Please send help.
7 miles
6 miles
5 miles
4 miles
3 miles
2 miles
1 mile
North (West)
South (East)
Goleta
Did the driver give enough
information to the operator?
No; should have given also
the direction (north or south
of Goleta?)
Jan. 2006
Global Positioning
Slide 5
A Highway Isn’t
a Straight Line
7 miles
Straight line distance
7 miles
Driving distance
Hello, 911? I have had an
accident on Highway 101,
seven miles from Goleta.
Please send an ambulance.
North (West)
South (East)
Goleta
Seven miles driving or
straight-line distance?
North or south of Goleta?
From now on, we speak of
straight-line distances only
Jan. 2006
Global Positioning
Slide 6
A More Complicated
Situation
Hello, 911? I have had an
accident on Highway 000,
seven miles east of Mytown.
Please send an ambulance.
7 miles
Straight line distance
4 miles
2 miles
West
East
000
Mytown
Did the driver give enough
information to the operator?
No – try to explain why
Remember all distances are
measured in straight lines
Jan. 2006
Global Positioning
Slide 7
Locating a Point in
Two Dimensions
Hello, 911? I was forced to
crash-land my private plane,
seven miles from Goleta.
Please send a helicopter.
7 miles
6 miles
5 miles
4 miles
3 miles
2 miles
1 mile
North (West)
South (East)
Goleta
Did the pilot give enough
information to the operator?
No; you cannot locate a
point on a 2D surface when
you know only one distance
Jan. 2006
Global Positioning
Slide 8
A Second Piece of
Information
7 miles
Hello, 911? I have crash-landed
my private plane, seven miles
from Goleta and six miles from
Santa Barbara. Send help ASAP.
Did the pilot
give enough
information
to the
operator?
6 miles
4 miles
2 miles
Goleta
Santa Barbara
Jan. 2006
Global Positioning
Slide 9
So, to Summarize . . .
Goleta
To locate a point in one dimension,
we need a distance and a direction
To locate a point in two dimensions,
we need two distances and a little bit
more information
Can you guess what we need to
locate a point in three dimensions?
Jan. 2006
Answer: Three distances, and a bit more
Global Positioning
Slide 10
Using Additional
Information
7 miles
Hello, 911? I have crash-landed my
private plane, seven miles from Goleta,
six miles from Santa Barbara, two miles
from Ranger Station X. Send help.
2 miles
Additional info
helps in
locating the
crash site
more
accurately
Ranger
Station X
6 miles
Goleta
Santa Barbara
Jan. 2006
Global Positioning
Slide 11
Activity 1: Locating a Point on a Curve
1. Draw a trail on the floor of a room (using a long piece of string) or on
the playground surface (using chalk). Name the two ends A and B.
A
F
B
L
E
G
J K
H
M
N
I
2. Mark ten or more points on the trail using pieces of masking tape.
3. Form two groups or teams: the lost party and the rescuers.
4. The lost party selects one of the points and measures the distances
from that point to one of the two endpoints of the trail.
5. The rescuers are given the distance and the endpoint from which
it was measured and they must find the lost party’s location.
Jan. 2006
Global Positioning
Slide 12
Activity 2: Locating a Point Inside a 2D Area
1. Consider a room, a basketball court, or another 2-dimensional area.
Name four of the corners A, B, C, D.
A
B
E H K M
F
I
N
G J L
C
D
2. Mark ten or so points on the floor using small pieces of masking tape.
3. Form two groups or teams: the lost party and the rescuers.
4. The lost party selects one of the points and measures the distances
from that point to each of the four corners A, B, C, and D.
5. The rescuers are given the four distances from the chosen point to
corners A, B, C, D and they must find the lost party’s location.
Challenge question: How many of the four distances are redundant?
Jan. 2006
Global Positioning
Slide 13
But How Do We Measure Distance?
You
Let a trained pigeon fly to Goleta and back,
time the round-trip, and multiply by speed
You
In GPS, we use radio signals, not pigeons.
Radio signals travel much much faster and
their round trip time is very hard to measure
Either way, the measurement is not totally precise.
When we say the distance to Goleta is 7 miles,
it can really be (for example) 6.9 miles to 7.1 miles
Jan. 2006
Global Positioning
Center
of Goleta
Center
of Goleta
Same thing happens if
you use a tape measure
or string that stretches
Slide 14
Activity 3: Measuring Distances
You want to measure the distance from your home to school in feet.
Perform as many of the following measurements as possible and
compare the results.
1. If you can walk to school, count the number of steps you take.
Then do several measurements of your step length (take 10 steps
and use a tape measure to determine the distance traveled).
2. Measure the time it takes to drive to school (e.g., 5 minutes),
estimate the average speed (e.g., 25 miles/hr), and do the math.
3. Use a car’s trip odometer to find the distance, and convert to feet.
4. Measure the length of the route on a map, that has a scale bar,
to compute the distance.
Challenge question: All the methods above give you the trip distance.
How would you go about determining the straight-line distance?
Jan. 2006
Global Positioning
Slide 15
Activity 4: Treasure Hunt
1. Consider a room, a playground, or any other area where you can hide
a small item (say, a marble). Name four spots in the area A, B, C, D.
2. Form two groups or teams: the treasure owners and treasure seekers.
3. The treasure owners hide their treasure so that it is difficult to find.
They then record the distances from the treasure to each of the four
spots A, B, C, D and write them down on a piece of paper.
4. Let’s pretend that the treasure seekers find the piece of paper many
years later, but discover that only the distance to point A is readable.
5. How can the treasure seekers go about finding the treasure?
6. Would the treasure seekers’ job become any easier if they knew that
the distances had been recorded from the largest to the smallest,
so that the distance to A is the largest of the four distances?
Jan. 2006
Global Positioning
Slide 16
Dealing with Approximate Distances
7 miles
In this case,
the emergency crew
must search an area
to find the plane
6.9 miles
Suppose 7 miles
really means 6.9 to
7.1 miles, because
our measurements
are not exact
6 miles
7.1 miles
Goleta
Santa Barbara
6.1 miles
5.9 miles
Jan. 2006
Global Positioning
Slide 17
How GPS Works: The Satellites
There are at least 24
satellites at any time;
new ones are launched
to replace older ones
that need to be retired
The satellites are
positioned so that from
any point on earth,
radio communication
with at least four of
them is possible
Each satellite follows a
single orbit, passing
over the same places
on earth each time
Jan. 2006
Global Positioning
Slide 18
How GPS Works: The Measurements
Satellites in known
positions
Approximate
location of
moving GPS
receiver
determined
from distance
to satellites
Fixed station
is used to
make the
location info
more precise
Communication
Moving GPS
receiver
Fixed
Station
Jan. 2006
Where
am I?
Global Positioning
Slide 19
Global Positioning Receivers
Notebook-size GPS receiver;
may be carried along or
mounted on a car’s dashboard
Pocket computer
or PDA with GPS
GPS-enabled
cell phone
Jan. 2006
Global Positioning
Wrist
watch
with
GPS
Slide 20
Next Lesson
February or March 2006
Jan. 2006
Global Positioning
Slide 21