1: GPS AMS 103, Spring 2017 Due: Thursday, February 9, 11:59 pm EST Quiz: Tuesday, February 14, in class Objectives To reinforce concepts relevant to GPS technology and the underlying mathematical concepts. To practice quantitative skills related to systems of equations and coordinate systems. Instructions List, on the first page, any students with whom you worked and/or any outside resources you used in preparing your solutions. You may not consult solutions to previous semesters' assignments. Clearly identify which problem you are solving at the start of the solution. Solutions must be clearly and logically presented. When applicable, show your work! The solution to each problem must be continuous; that is, parts should not be crossed out and solutions to other problems should not be interspersed. Your solutions must be submitted through Blackboard as a single PDF file. Improperly submitted or late homework will not receive credit. Grading Policy This assignment will be graded for both completeness and accuracy. Completion of the assignmentdid you attempt every problem?will be worth 50% of the score. The problems will also be graded for accuracy; that is, did you provide a correct answer and, when applicable, show your work? Accuracy will determine the remaining 50%. Remember, if you use Wolfram Alpha (or another software tool), include a screen shot with both input and pertinent output to show your work. Also, wikipedia is not a valid source (although it may be a great starting point to find more acceptable sources!). Problems 1. (?, 16 pts) How does a GPS receiver know the locations of the satellites when determining its location? Hint: The Trimble tutorial might be useful. 2. (??, 26 pts) Meteorites regularly enter the atmosphere, where they heat up, disintegrate, and finally explode before hitting the surface of the Earth. Such an explosion generates a shock wave (traveling at the speed of sound) that can be detected by seismographs installed at various locations on the Earth's surface. Each seismograph (equipped with a perfectly synchronized clock) then determines the time it took the shockwave to reach the seismograph. How many seismographs need to detect the shock wave to determine the location of the meteorite explosion? 3. (?, 16 pts) For each of the following systems of equations, (i) determine the number of equations and the number of unknown variables in the system and (ii) classify the system as consistent or inconsistent. If the system is consistent, (iii) solve the system. Homework 1: GPS Page 2/2 Hint: items (ii) and (iii) are closely related; Wolfram Alpha may be useful. (a) |x + 7| = 9, |x + 1| = 3, |x 4| = 2. (b) p 2q = 5, p + 2q = 1, p q = 3. 4. (?, 16 pts) Calculate the latitude and longitude angles for the following Cartesian coordinates. (a) (x = 2, y = 7, z = 1). (b) (x = 5, y = 1, z = 3). 5. (??, 26 pts) The following table shows the locations of satellites and their distances from you. All coordinates and distances are in miles. Satellite x y z 1 4608.6 6581.76 9575.56 2 2162.34 12263.3 1089.45 3 9430.08 7912.78 2170.6 Distance 8700.78 11447.6 11568.6 (a) Determine if each of the following points could be your location. Justify why or why not. (b) If you know you are also on the surface of the Earth (assume radius is 4000 mi), what is your location? If unknown, explain why. (i) (x = 476.824, y = 2065.35, z = 3392.19). (ii) (x = 2264.84, y = 11859.9, z = 10350.6). AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Global Positioning System (GPS) Location, Location, Location Mathematics of GPS Matthew G. Reuter AMS 103: Applied Mathematics in Modern Technology Stony Brook University Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together {January 26, 31, February 2}, 2017 1 / 58 Warmup Puzzle AMS 103: Global Positioning System Matt Reuter Not graded, for your enjoyment only! Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together From http://www.puzzle.dse.nl/math/index us.html 2 / 58 Agenda AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Recap: What is Applied Math? Location, Location, Location Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Putting it All Together 3 / 58 What is Applied Mathematics? AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together http://xkcd.com/435/ 4 / 58 What is Applied Mathematics? AMS 103: Global Positioning System Matt Reuter Quoting Prager (1972, see The Princeton Companion to Applied Mathematics) Precisely to define applied mathematics is next to impossible. It cannot be done in terms of subject matter . . . [nor] can it be done in terms of motivation. . . . Perhaps the best I can do . . . is to describe applied mathematics as the bridge connecting pure mathematics with science and technology. Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together http://press.princeton.edu/titles/10592.html 5 / 58 AMS 103: Global Positioning System The Goal of AMS 103 Matt Reuter Explore various technologies with an emphasis on their mathematical underpinnings. Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Mathematics Technology This is not an engineering course; we won't go into every detail of a given technology. http://www.merton.ox.ac.uk/node/139 https://www.youtube.com/watch?v=Pk1ue1tolFc 6 / 58 Use Technology to Inspire/Motivate Mathematics AMS 103: Global Positioning System Matt Reuter Our Approach Recap: What is Applied Math? 1. Discuss a technology; its uses; its goals Location, Location, Location 2. Ask a question (or two) related to the technology Mathematics of GPS 3. Identify mathematical challenge(s)/concept(s) needed for addressing our question; analyze 4. Develop a solution strategy; demonstrate with examples 5. Answer our technological question, if possible Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together We use this approach to develop critical analysis skills. That is, why do we discuss the mathematical concepts? What is our rationale? So, let's dive in to our first technology. 7 / 58 Where are you? AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 8 / 58 Where are you? AMS 103: Global Positioning System Matt Reuter An AMS 103 lecture? Stony Brook University? Recap: What is Applied Math? Location, Location, Location How do you know? These questions all get at localization; that is, methods for determining location. Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Historical Approaches 8 / 58 Where are you? AMS 103: Global Positioning System Matt Reuter An AMS 103 lecture? Stony Brook University? Recap: What is Applied Math? Location, Location, Location How do you know? These questions all get at localization; that is, methods for determining location. Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Historical Approaches I Landmarks (if available); can be natural or human-made I Stars and constellations (at night); see information on sextants or celestial navigation 8 / 58 Modern Localization: The Global Positioning System (GPS) Online Tutorial (Outside Reading) There is a phenomenal tutorial (with animations!) at Trimble. http://www.trimble.com/gps tutorial/ I The GPS is a network of ground stations and a constellation of 24 (or more, including backups) satellites in the Earth's orbit. I The GPS satellites serve as \"human-made stars\" that can be used as known \"celestial bodies\". Their locations are known at any given moment. I Advanced forms of GPS can provide resolution better than 1 centimeter (ethical issues notwithstanding). AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together http://www.trimble.com/gps tutorial/whatgps.aspx 9 / 58 Why GPS? I I I Previous technologies for localization had limited ranges of operation (landmarks, LORAN) or times of operation (celestial navigation), were prone to errors (dead reckoning), had poor resolution, etc. In the Cold War, the U.S. arsenal was predominantly ship-based; the military needed to accurately know mobile locations, should missiles be needed. U.S. Department of Defense spent $12 billion to develop, deploy, and validate the constellation of satellites. Work completed in 1995, but began in 1973. AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together https://en.wikipedia.org/wiki/Global Positioning System 10 / 58 How Does GPS Work? AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? At any time, a GPS receiver (on the ground) measures radio signals sent from satellites in orbit. From these measurements, location can be accurately calculated. Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Picture from Trimble 11 / 58 How Does GPS Work? (More Details) AMS 103: Global Positioning System Matt Reuter GPS has five key steps: 1. The basis of GPS is \"triangulation\" from satellites. 2. To \"triangulate,\" a GPS receiver measures distance using the travel time of radio signals. 3. To measure travel time, GPS needs very accurate timing which it achieves with some tricks. 4. Along with distance, you need to know exactly where the satellites are in space. High orbits and careful monitoring are the secret. Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 5. Finally you must correct for any delays the signal experiences as it travels through the atmosphere. http://www.trimble.com/gps tutorial/howgps.aspx 12 / 58 Mathematics of GPS AMS 103: Global Positioning System Matt Reuter Our Driving Question How many satellites' signals do we need to determine our location? Let's tackle this question by building a model step-by-step. What mathematical concepts are needed? (Okay, one of them is physical. . . ) I Relationship between speed, distance, and time I Triangulation (maybe we mean \"trilateration\"?) I Systems of equations I Coordinate systems Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 13 / 58 Speed and Distance (or velocity and displacement if you prefer vectors. . . ) We need to know how long it takes a signal from a GPS satellite to reach our receiver. AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together http://competitionzenith.blogspot.com/2015/03/speedtime-and-distance-std.html s= d d or d = s t or t = t s Note that this is a physics concept, not a math concept! 14 / 58 Speed and Distance Examples AMS 103: Global Positioning System Matt Reuter Example 1 It takes you 10 minutes to drive from Stony Brook University to Smith Haven Mall. What is your average speed? Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 15 / 58 AMS 103: Global Positioning System Speed and Distance Examples Matt Reuter Example 1 It takes you 10 minutes to drive from Stony Brook University to Smith Haven Mall. What is your average speed? Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Answer: 19.8 mph Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 15 / 58 AMS 103: Global Positioning System Speed and Distance Examples Matt Reuter Example 1 It takes you 10 minutes to drive from Stony Brook University to Smith Haven Mall. What is your average speed? Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Answer: 19.8 mph Example 2 (Maybe a bit more relevant) Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together A satellite orbits the Earth at an altitude of 11,000 miles and emits a radio signal (at the speed of light). How long does it take the signal to reach the point on the Earth directly \"below\" the satellite? 15 / 58 AMS 103: Global Positioning System Speed and Distance Examples Matt Reuter Example 1 It takes you 10 minutes to drive from Stony Brook University to Smith Haven Mall. What is your average speed? Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Answer: 19.8 mph Example 2 (Maybe a bit more relevant) Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together A satellite orbits the Earth at an altitude of 11,000 miles and emits a radio signal (at the speed of light). How long does it take the signal to reach the point on the Earth directly \"below\" the satellite? Answer: 0.06 s Our GPS satellites and receivers need accurate clocks. 15 / 58 Distance Between Two Points AMS 103: Global Positioning System Matt Reuter Given two points in space, what is the distance between them? (Or, how far is a GPS satellite from our receiver?) q I 1-D: d = (x1 x2 )2 = |x1 x2 | q I 2-D: d = (x1 x2 )2 + (y1 y2 )2 q I 3-D: d = (x1 x2 )2 + (y1 y2 )2 + (z1 z2 )2 Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Example What is the distance between the points (3, 2, 1) and (2, 2, 13)? 16 / 58 AMS 103: Global Positioning System Distance Between Two Points Matt Reuter Given two points in space, what is the distance between them? (Or, how far is a GPS satellite from our receiver?) q I 1-D: d = (x1 x2 )2 = |x1 x2 | q I 2-D: d = (x1 x2 )2 + (y1 y2 )2 q I 3-D: d = (x1 x2 )2 + (y1 y2 )2 + (z1 z2 )2 Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Example What is the distance between the points (3, 2, 1) and (2, 2, 13)? Answer: 13 units 16 / 58 Triangulation (A Math Concept!) Definition (from wikipedia) Triangulation is \"the process of determining the location of a point by measuring angles to it from known points at either end of fixed baseline.\" AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Historically used in surveying https://en.wikipedia.org/wiki/Triangulation 17 / 58 Trilateration Are we really triangulating for GPS? AMS 103: Global Positioning System Matt Reuter Definition (again from wikipedia) Recap: What is Applied Math? Trilateration is \"the process of determining absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres, or triangles.\" Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together http://moorefarmsbg.org/signals-from-space/gps 3d-trilateration/ 18 / 58 Trilateration from Satellites AMS 103: Global Positioning System Matt Reuter Suppose we know the time it takes for a GPS satellite's signal to reach us. Can we determine our position? Recap: What is Applied Math? Location, Location, Location Let's work our way up to this question. 1-D Example (start simple); numbers are not realistic Suppose a satellite is positioned at the origin of a 1-D line. It emits a signal that travels at 4 miles per second. You receive the signal 3 seconds later. What is your position? Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together https://en.wikipedia.org/wiki/Coordinate system 19 / 58 AMS 103: Global Positioning System Trilateration from Satellites Matt Reuter Suppose we know the time it takes for a GPS satellite's signal to reach us. Can we determine our position? Recap: What is Applied Math? Location, Location, Location Let's work our way up to this question. Mathematics of GPS 1-D Example (start simple); numbers are not realistic Suppose a satellite is positioned at the origin of a 1-D line. It emits a signal that travels at 4 miles per second. You receive the signal 3 seconds later. What is your position? Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Answer: Inconclusive https://en.wikipedia.org/wiki/Coordinate system 19 / 58 Trilateration from Satellites (2) A second satellite? Suppose, in addition to before, there is a second satellite positioned at 4 miles. It emits a signal (that travels at the same speed) at the same time as the first satellite. You receive this signal after 2 seconds. What is your position? AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 20 / 58 AMS 103: Global Positioning System Trilateration from Satellites (2) Matt Reuter A second satellite? Suppose, in addition to before, there is a second satellite positioned at 4 miles. It emits a signal (that travels at the same speed) at the same time as the first satellite. You receive this signal after 2 seconds. What is your position? Answer: 12 miles We need at least two satellites (in this 1-D example) to determine our location! Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 20 / 58 AMS 103: Global Positioning System Trilateration from Satellites (2) Matt Reuter A second satellite? Suppose, in addition to before, there is a second satellite positioned at 4 miles. It emits a signal (that travels at the same speed) at the same time as the first satellite. You receive this signal after 2 seconds. What is your position? Answer: 12 miles We need at least two satellites (in this 1-D example) to determine our location! Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together What if. . . . . . the second satellite was positioned at 3 miles and everything else remained the same? What is your position? 20 / 58 AMS 103: Global Positioning System Trilateration from Satellites (2) Matt Reuter A second satellite? Suppose, in addition to before, there is a second satellite positioned at 4 miles. It emits a signal (that travels at the same speed) at the same time as the first satellite. You receive this signal after 2 seconds. What is your position? Answer: 12 miles We need at least two satellites (in this 1-D example) to determine our location! Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together What if. . . . . . the second satellite was positioned at 3 miles and everything else remained the same? What is your position? Answer: Something's wrong 20 / 58 Can We Find a Mathematical Structure? AMS 103: Global Positioning System Matt Reuter Before we make the examples more complicated (2-D or 3-D cases), let's take a step back and analyze the problems we just solved. The examples with two satellites gave us the equations q st1 = (x x1 )2 q st2 = (x x2 )2 , Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together where s is the signal speed, t1(2) is the time it took for us to receive the signal from satellite 1(2), x1(2) is the position of satellite 1(2), and x is our position. This is a system of two equations. We are looking for a solution x that simultaneously satisfies both equations. 21 / 58 Systems of Equations AMS 103: Global Positioning System Matt Reuter Definition A system of equations is a set of equations that involve the same unknown variable(s). A solution provides value(s) for the unknown variable(s) that simultaneously satisfy all equations in the system. From our previous examples on GPS, the system was p p st1 = (x x1 )2 and st2 = (x x2 )2 Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 2 equations, 1 unknown variable (x). Our location can be any value x that satisfies all the equations. A good rule of thumb (but by no means a rigorous statement) is that we want one equation for each unknown variable. 22 / 58 Characterizing a System of Equations Language is a powerful tool! I The set of all solutions is called the solution set. I A system may not have a solution (see the \"What if. . . \" example earlier). Such a system is called inconsistent. I If a system has a solution, it is consistent. I A solution, if it exists, does not need to be unique. That is, there may be 2, 3, 4, . . . , or even an infinite number of solutions to the system. AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 23 / 58 Characterizing a System of Equations Language is a powerful tool! I The set of all solutions is called the solution set. I A system may not have a solution (see the \"What if. . . \" example earlier). Such a system is called inconsistent. I If a system has a solution, it is consistent. I A solution, if it exists, does not need to be unique. That is, there may be 2, 3, 4, . . . , or even an infinite number of solutions to the system. AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together For our ongoing GPS question (How many satellites do we need?), we're really looking for the minimal number of signals (equations) that leads to a unique solution. Of course, each equation has a specific form in this case: speed transit time = distance from our position 23 / 58 Example: Solving a System of Equations Consider the system we saw in the previous examples: p p st1 = (x x1 )2 and st2 = (x x2 )2 We can algebraically manipulate these equations to produce x= s 2 (t12 t22 ) + x22 x12 2(x2 x1 ) (of course, x1 6= x2 ; the satellites are at different places). AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together But. . . This suggests that there is always a solution (assuming x1 6= x2 ). We saw an example earlier where the system was inconsistent (dropping units, s = 4, t1 = 3, x1 = 0, t2 = 2, x2 = 3). How is this possible? 24 / 58 Example: Solving a System of Equations Consider the system we saw in the previous examples: p p st1 = (x x1 )2 and st2 = (x x2 )2 We can algebraically manipulate these equations to produce x= s 2 (t12 t22 ) + x22 x12 2(x2 x1 ) (of course, x1 6= x2 ; the satellites are at different places). AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together But. . . This suggests that there is always a solution (assuming x1 6= x2 ). We saw an example earlier where the system was inconsistent (dropping units, s = 4, t1 = 3, x1 = 0, t2 = 2, x2 = 3). How is this possible? Pro-tip: Verify your solution! 24 / 58 Checking Your Work AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together https://xkcd.com/809/ 25 / 58 Tips for Solving a System of Equations AMS 103: Global Positioning System Matt Reuter In general, there is no recipe. Strategies to consider: I Solve one equation for one of the unknowns and substitute this expression into the other equations. This eliminates one unknown, (hopefully) making the system simpler. I Multiply both sides of one equation by the same non-zero constant. I Add (or subtract) one equation from another. Combined with the previous step, this can perhaps eliminate certain terms. I Square both sides of one equation. Careful! This may lead to invalid solutions. Verifying your solution is a must if you do this. Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Which of these techniques were used in our example? 26 / 58 Additional Resources AMS 103: Global Positioning System Matt Reuter There are lots of resources online for help on solving systems of equations. Most focus on systems of linear equations, but some are general. I Khan Academy (link) I Wolfram Alpha (link). Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Interested in More Information? Linear algebra courses (e.g., AMS 210) spend quite a bit of time on techniques for solving systems of linear equations. 27 / 58 Output from the Wolfram Alpha Query AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 28 / 58 A Note on Using Wolfram Alpha AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? I fully encourage you to use Wolfram Alpha (and other tools) on the web to help on the homework. However, \"Wolfram Alpha \" alone is not sufficient for showing your work. Please include a screenshot of your input and the pertinent output. Remember also that you will not have access to online tools during the quizzes and exam. Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together For the purpose of this unit's quiz, you should know how to verify the solution of a system of equations. 29 / 58 Graphically Representing a System of Equations The Wolfram Alpha example showed that a system of equations can be viewed graphically. Let's develop this more. From our first trilateration example (d = 12 mi): AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Inconclusive location (two possibilities)! 30 / 58 Graphics for Two Satellite Measurements (1D) What about the second trilateration example? AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together A unique solution, as expected! 31 / 58 An Inconsistent System (1D) AMS 103: Global Positioning System Matt Reuter What about the third trilateration example? Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together No solutions. 32 / 58 Adding Realism: Two Dimensions! AMS 103: Global Positioning System Matt Reuter All examples so far have considered 1-D problems. 1 satellite signal narrowed our location to 2 places and a second signal excluded one (or indicated something's wrong). A Two-Dimensional Example (Unrealistic Numbers) A satellite is positioned at (2, 3) miles and emits a signal that you receive 3 seconds later. The signal travels at 1 mile/second. What is your location? Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 33 / 58 Adding Realism: Two Dimensions! AMS 103: Global Positioning System Matt Reuter All examples so far have considered 1-D problems. 1 satellite signal narrowed our location to 2 places and a second signal excluded one (or indicated something's wrong). A Two-Dimensional Example (Unrealistic Numbers) A satellite is positioned at (2, 3) miles and emits a signal that you receive 3 seconds later. The signal travels at 1 mile/second. What is your location? Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Answer: Inconclusive ( possibilities) 33 / 58 Graphical Representation in Two Dimensions AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 34 / 58 AMS 103: Global Positioning System Add a Satellite Matt Reuter Two Satellites Satellites synchronously emit signals that travel at 1 mi/s. Satellite 1 2 Position (miles) (2, 3) (1, 1) What's your position? Transit Time (seconds) 3 4 Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 35 / 58 AMS 103: Global Positioning System Add a Satellite Matt Reuter Two Satellites Satellites synchronously emit signals that travel at 1 mi/s. Satellite 1 2 Position (miles) (2, 3) (1, 1) Transit Time (seconds) 3 4 What's your position? Answer: Still inconclusive (but down to 2) Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 35 / 58 AMS 103: Global Positioning System Add One More Satellite Matt Reuter Three Satellites Satellites synchronously emit signals that travel at 1 mi/s. Satellite 1 2 3 Position (miles) (2, 3) (1, 1) (5, 6) What's your position? Transit Time (seconds) 3 4 5 Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together 36 / 58 AMS 103: Global Positioning System Add One More Satellite Matt Reuter Three Satellites Satellites synchronously emit signals that travel at 1 mi/s. Satellite 1 2 3 Position (miles) (2, 3) (1, 1) (5, 6) Transit Time (seconds) 3 4 5 What's your position? Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together Answer: (1, 3) miles 36 / 58 Summary of Trilateration Results So what can we determine from some number of satellite signals? (Assume GPS is working properly.) # Signals 1 2 3 4 1-D 2 Points Location! 2-D On a circle 2 Points Location! AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together a 37 / 58 AMS 103: Global Positioning System Summary of Trilateration Results So what can we determine from some number of satellite signals? (Assume GPS is working properly.) # Signals 1 2 3 4 1-D 2 Points Location! 2-D On a circle 2 Points Location! 3-D On a sphere Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together https://en.wikipedia.org/wiki/Sphere 37 / 58 AMS 103: Global Positioning System Summary of Trilateration Results So what can we determine from some number of satellite signals? (Assume GPS is working properly.) # Signals 1 2 3 4 1-D 2 Points Location! 2-D On a circle 2 Points Location! 3-D On a sphere On a circle Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together http://www.mathalino.com/reviewer/solid-mensuration-solid-geometry/sphere 37 / 58 AMS 103: Global Positioning System Summary of Trilateration Results So what can we determine from some number of satellite signals? (Assume GPS is working properly.) # Signals 1 2 3 4 1-D 2 Points Location! 2-D On a circle 2 Points Location! 3-D On a sphere On a circle 2 Points Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together https://commons.wikimedia.org/wiki/File:Sphere3-intersect.svg 37 / 58 AMS 103: Global Positioning System Summary of Trilateration Results So what can we determine from some number of satellite signals? (Assume GPS is working properly.) # Signals 1 2 3 4 1-D 2 Points Location! 2-D On a circle 2 Points Location! 3-D On a sphere On a circle 2 Points Location! Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together http://moorefarmsbg.org/signals-from-space/gps 3d-trilateration/ 37 / 58 Living in the Real World Take a closer look at that last picture. AMS 103: Global Positioning System Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together http://moorefarmsbg.org/signals-from-space/gps 3d-trilateration/ Counting is Fun How many spheres do you see? 38 / 58 AMS 103: Global Positioning System Living in the Real World Take a closer look at that last picture. Matt Reuter Recap: What is Applied Math? Location, Location, Location Mathematics of GPS Speed and Distance Trilateration Systems of Equations Trilateration (II) Coordinate Systems Putting it All Together http://moorefarmsbg.org/signals-from-space/gps 3d-trilateration/ Counting is Fun How many spheres do you see? Answer: 5 So why do we need four satellite signals? The Earth should count as one of the spheres. . . 38 / 58 AMS 103: Global Positioning System Synchronization Matt Reuter (not to be confused with \"Synchronicity\