jan. 2006global positioningslide 1 global positioning a lesson in the “math + fun!” series

Jan. 2006 Global Positioning Slide 1

Global PositioningA Lesson in the “Math + Fun!” Series


About This Presentation

Edition Released Revised Revised

First Jan. 2006

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


We Need to Know Where We AreA ship at sea may be lost when we can’t determine its location

A plane flying over cloud-covered mountains needs location info


GPS Provides Location Information

For navigation:Hiking,Driving,Flying,Sailing

For assistance: Rescue,The visually impaired

For tracking: Earth,Structures,Children,Autos, Pets,Wildlife

For recreation: Gaming,Dining,Sports


Locating a Point in One Dimension

Goleta

1 mile

5 miles

3 miles

2 miles

4 miles

6 miles

7 miles

Hello, 911? I have had an accident on Highway 101, seven miles from Goleta. Please send help.

Did the driver give enough information to the operator?

North (West)

South (East)

No; should have given also the direction (north or south of Goleta?)


A Highway Isn’t a Straight Line

Goleta

Hello, 911? I have had an accident on Highway 101, seven miles from Goleta. Please send an ambulance.

Seven miles driving or straight-line distance?

North (West)

South (East)

North or south of Goleta?

7 milesDriving distance

7 milesStraight line distance

From now on, we speak of straight-line distances only


A More Complicated Situation

Mytown

Hello, 911? I have had an accident on Highway 000, seven miles east of Mytown. Please send an ambulance.

Did the driver give enough information to the operator?

West

East

No – try to explain why

7 milesStraight line distance

Remember all distances are measured in straight lines

000

4 miles

2 miles


Locating a Point in Two Dimensions

Goleta

1 mile

5 miles

3 miles

2 miles

4 miles

6 miles

7 miles

Hello, 911? I was forced to crash-land my private plane, seven miles from Goleta. Please send a helicopter.

Did the pilot give enough information to the operator?

North (West)

South (East)

No; you cannot locate a point on a 2D surface when you know only one distance


A Second Piece of Information

Goleta

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?

Santa Barbara

2 miles

4 miles

6 miles


So, to Summarize . . .

To locate a point in one dimension, we need a distance and a direction

Goleta

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? Answer: Three distances, and a bit more


Using Additional Information

Goleta

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.

Additional info helps in locating the crash site more accurately

Santa Barbara

2 miles

6 miles

Ranger Station X


Activity 1: Locating a Point on a Curve1. 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.

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.

AB

E

IH

G K N

F

J ML


Activity 2: Locating a Point Inside a 2D Area1. Consider a room, a basketball court, or another 2-dimensional area. Name four of the corners A, B, 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.

A B

C D

E

IH

G

K

NFJ

M

L

Challenge question: How many of the four distances are redundant?


But How Do We Measure Distance?

Center of Goleta

You

Center of Goleta

You

Let a trained pigeon fly to Goleta and back, time the round-trip, and multiply by speed

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

Same thing happens if you use a tape measure or string that stretches


Activity 3: Measuring DistancesYou 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?


Activity 4: Treasure Hunt1. 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.

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.

2. Form two groups or teams: the treasure owners and treasure seekers.

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?


Dealing with Approximate Distances

Goleta

7 miles

In this case, the emergency crew must search an area to find the plane

Santa Barbara

6 miles

6.9 miles

7.1 miles

5.9 miles

6.1 miles

Suppose 7 miles really means 6.9 to 7.1 miles, because our measurements are not exact


How GPS Works: The Satellites

The satellites are positioned so that from any point on earth, radio communication with at least four of them is possible

There are at least 24 satellites at any time; new ones are launched to replace older ones that need to be retired

Each satellite follows a single orbit, passing over the same places on earth each time


How GPS Works: The MeasurementsSatellites in known

positions

Moving GPS receiver

FixedStation

Communication

Approximate location of moving GPS receiver determined from distance to satellites

Fixed station is used to make the location info more precise

Where am I?


Global Positioning Receivers

Notebook-size GPS receiver;may be carried along or mounted on a car’s dashboard

GPS-enabled cell phone

Pocket computer or PDA with GPS

Wrist watch with GPS


Next LessonFebruary or March 2006

jan. 2006global positioningslide 1 global positioning a lesson in the “math + fun!” series

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