1. Jan. 2006Global PositioningSlide 1 Global Positioning A Lesson in the “Math + Fun!” Series

2. Jan. 2006Global PositioningSlide 2 About This Presentation EditionReleasedRevised FirstJan. 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

3. Jan. 2006Global PositioningSlide 3 We Need to Know Where We Are A ship at sea may be lost when we can’t determine its location A plane flying over cloud-covered mountains needs location info

4. Jan. 2006Global PositioningSlide 4 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

5. Jan. 2006Global PositioningSlide 5 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?)

6. Jan. 2006Global PositioningSlide 6 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 miles Driving distance 7 miles Straight line distance From now on, we speak of straight-line distances only

7. Jan. 2006Global PositioningSlide 7 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 miles Straight line distance Remember all distances are measured in straight lines 000 4 miles 2 miles

8. Jan. 2006Global PositioningSlide 8 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

9. Jan. 2006Global PositioningSlide 9 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

10. Jan. 2006Global PositioningSlide 10 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

11. Jan. 2006Global PositioningSlide 11 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

12. Jan. 2006Global PositioningSlide 12 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. 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. A B E I H GK N F J M L

13. Jan. 2006Global PositioningSlide 13 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. 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. AB CD E I H G K N F J M L Challenge question: How many of the four distances are redundant?

14. Jan. 2006Global PositioningSlide 14 But How Do We Measure Distance? Center of Goleta YouCenter 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

15. Jan. 2006Global PositioningSlide 15 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?

16. Jan. 2006Global PositioningSlide 16 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. 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?

17. Jan. 2006Global PositioningSlide 17 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

18. Jan. 2006Global PositioningSlide 18 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

19. Jan. 2006Global PositioningSlide 19 How GPS Works: The Measurements Satellites in known positions Moving GPS receiver Fixed Station 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?

20. Jan. 2006Global PositioningSlide 20 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

21. Jan. 2006Global PositioningSlide 21 Next Lesson February or March 2006