Gps

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Satellite frequencies
All satellites broadcast at the same two frequencies, 1.57542 GHz (L1 signal) and 1.2276 GHz (L2 signal). The satellite network uses a CDMA spread-spectrum technique where the lowbitrate message data is encoded with a high-rate pseudo-random (PRN) sequence that is different for each satellite. The receiver must be aware of the PRN codes for each satellite to reconstruct the actual message data. The C/A code, for civilian use, transmits data at 1.023 million chips per second, whereas the P code, for U.S. military use, transmits at 10.23 million chips per second. The actual internal reference of the satellites is 10.22999999543 MHz to compensate for relativistic effects[73] [74] that make observers on Earth perceive a different time reference with respect to the transmitters in orbit. The L1 carrier is modulated by both the C/A and P codes, while the L2 carrier is only modulated by the P code.[75] The P code can be encrypted as a so-called P(Y) code that is only available to military equipment with a proper decryption key. Both the C/A and P(Y) codes impart the precise time-of-day to the user. The L3 signal at a frequency of 1.38105 GHz is used to transmit data from the satellites to ground stations. This data is used by the United States Nuclear Detonation (NUDET) Detection System (USNDS) to detect, locate, and report nuclear detonations (NUDETs) in the Earth's atmosphere and near space.[76] One usage is the enforcement of nuclear test ban treaties. The L4 band at 1.379913 GHz is being studied for additional ionospheric correction.[citation needed] The L5 frequency band at 1.17645 GHz was added in the process of GPS modernization. This frequency falls into an internationally protected range for aeronautical navigation, promising little or no interference under all circumstances. The first Block IIF satellite that would provide this signal is set to be launched in 2009.[77] The L5 consists of two carrier components that are in phase quadrature with each other. Each carrier component is bi-phase shift key (BPSK) modulated by a separate bit train. "L5, the third civil GPS signal, will eventually support safetyof-life applications for aviation and provide improved availability and accuracy."[78] A conditional waiver has recently been granted to LightSquared to operate a terrestrial broadband service near the L1 band. Although LightSquared had applied for a license to operate in the 1525 to 1559 band as early as 2003 and it was put out for public comment, the FCC asked LightSquared to form a study group with the GPS community to test GPS receivers and identify issue that might arise due to the larger signal power from the LightSquared terrestrial network. The GPS community had not objected to the LightSquared (formerly MSV and SkyTerra) applications until November 2010, when LightSquared applied for a modification to its Ancillary Terrestrial Component (ATC) authorization. This filing (SAT-MOD-20101118-00239) amounted to a request to run several orders of magnitude more power in the same frequency band for terrestrial base stations, essentially repurposing what was supposed to be a "quiet neighborhood" for signals from space as the equivalent of a cellular network. Testing in the first half of 2011 has demonstrated that the impact of the lower 10 MHz of spectrum is minimal to GPS devices (less than 1% of the total GPS devices are affected). The upper 10 MHz intended for use by LightSquared may have some impact on GPS devices. There is some concern that this

will seriously degrade the GPS signal for many consumer uses.[79][80] Aviation Week magazine reports that the latest testing (June 2011) confirms "significant jamming" of GPS by LightSquared's system.

Introduction
Trying to figure out where you are is probably one of humankind's oldest problems. Navigation and positioning are crucial to so many activities and yet the process has always been quite cumbersome and inexact. In the earliest days mankind used the stars to navigate. Early instruments also sited the stars to determine position. The science of horology began in part because navigation depended on precise timing the movement of the stars. Over the years all kinds of technologies have tried to simplify the task but every one has had some disadvantage. Finally, the U.S. Department of Defense decided that the military had to have a precise form of worldwide positioning. Fortunately they had the deep pockets it took to build something really good. The result is the Global Positioning System, a system that's changed navigation forever. The Global Positioning System (GPS) is a worldwide radio-navigation system formed from a constellation of 24 satellites and their ground stations. GPS uses these "man-made stars" as reference points to calculate positions accurate to a matter of meters. In fact, with advanced forms of GPS you can make measurements to better than a centimeter! In a sense it's like giving every square meter on the planet a unique address. GPS receivers have been miniaturized to just a few integrated circuits and so are becoming very economical. And that makes the technology accessible to virtually everyone. These days GPS is finding its way into cars, boats, planes, construction equipment, movie making gear, farm machinery, even laptop computers. Soon GPS will become almost as basic as the cell telephone. In fact, GPS technology is being used now in cell phones to implement the emergency ('911') location system.

How GPS Works
Here's how GPS works in six logical 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. 5. You must correct for any delays the signal experiences as it travels through the atmosphere. 6. Finally (for us), you can now obtain the precise time from the GPS satellites. We'll explain each of these points in subsequent sections. Improbable as it may seem, the whole idea behind GPS is to use satellites in space as reference points for locations here on earth. That's right, by very, very accurately measuring our distance from three satellites we can "triangulate" our position anywhere on earth. Forget for a moment how our receiver measures this distance. We'll get to that later. First consider how distance measurements from three satellites can pinpoint you in space.

Triangulation
    Position is calculated from distance measurements (ranges) to satellites. Mathematically we need four satellite ranges to determine exact position. Three ranges are enough if we reject ridiculous answers or use other tricks. Another range is required for technical reasons to be discussed later.

The Geometry: Suppose we measure our distance from a satellite and find it to be 10,000 miles. Knowing that we're 10,000 miles from a particular satellite narrows down all the possible locations we could be in the whole universe to the surface of a sphere that is centered on this satellite and has a radius of 10,000 miles Next, say we measure our distance to a second satellite and find out that it's 11,000 miles away. That tells us that we're not only on the first sphere but we're also on a sphere that's 11,000 miles from the second satellite. Or in other words, we're somewhere on the circle where these two spheres intersect.

If we then make a measurement from a third satellite and find that we're 12,000 miles from that one, that narrows our position down even further, to the two points where the 12,000 mile sphere cuts through the circle that's the intersection of the first two spheres. So by ranging from three satellites we can narrow our position to just two points in space. This arrangement of satellites is also called a "constellation". To decide which one is our true location we could make a fourth measurement. But usually one of the two points is a ridiculous answer (either too far from Earth or moving at an impossible velocity) and can be rejected without a measurement.

A fourth measurement does come in very handy for another reason however, but we'll tell you about that later. Next we'll see how the system measures distances to satellites.

Measuring Distance
1. Distance to a satellite is determined by measuring how long a radio signal takes to reach us from that satellite. 2. To make the measurement we assume that both the satellite and our receiver are generating the same pseudo-random codes at exactly the same time. 3. By comparing how late the satellite's pseudo-random code appears compared to our receiver's code, we determine how long it took to reach us. 4. Multiply that travel time by the speed of light and you've got distance. Discussion We saw in the last section that a position is calculated from distance measurements to at least three satellites. But how can you measure the distance to something that's floating around in space? We do it by timing how long it takes for a signal sent from the satellite to arrive at our receiver.

The Math In a sense, the whole thing boils down to those "velocity times travel time" math problems we did in high school. Remember the old: "If a car goes 60 miles per hour for two hours, how far does it travel?"
Velocity (60 mph) x Time (2 hours) = Distance (120 miles)

In the case of GPS we're measuring a radio signal so the velocity is going to be the speed of light or roughly 186,000 miles per second. The problem is measuring the travel time

The timing problem is tricky. First, the times are going to be awfully short. If a satellite were right overhead the travel time would be something like 0.06 seconds. So we're going to need some really precise clocks. We'll talk about those soon. But assuming we have precise clocks, how do we measure travel time? To explain it let's use a goofy analogy: Suppose there was a way to get both the satellite and the receiver to start playing "Stairway to Heaven" at precisely 12 noon. If sound could reach us from space then standing at the receiver we'd hear two versions of the 'Stairway to Heaven', one from our

receiver and one from the satellite. These two versions would be out of sync. The version coming from the satellite would be a little delayed because it had to travel more than 11,000 miles. If we wanted to see just how delayed the satellite's version was, we could start delaying the receiver's version until they fell into perfect sync. The amount we have to shift back the receiver's version is equal to the travel time of the satellite's version. So we just multiply that time times the speed of light and voila! we've got our distance to the satellite. That's basically how GPS works. Only instead of 'Stairway to Heaven' the satellites and receivers use something called a "Pseudo Random Code" - which is probably quicker to sing than 'Stairway to Heaven'. Random Code The Pseudo Random Code (PRC) is a fundamental part of GPS. Physically it's just a very complicated digital code, or in other words, a complicated sequence of "on" and "off" pulses. The signal is so complicated that it almost looks like random electrical noise. Hence the name "Pseudo-Random." There are several good reasons for that complexity: First, the complex pattern helps make sure that the receiver doesn't accidentally sync up to some other signal. The patterns are so complex that it's highly unlikely that a stray signal will have exactly the same shape. Since each satellite has its own unique Pseudo-Random Code this complexity also guarantees that the receiver won't accidentally pick up another satellite's signal. So all the satellites can use the same frequency without jamming each other. And it makes it more difficult for a hostile force to jam the system. In fact the Pseudo Random Code gives the Department of Defense a way to control access to the system. But there's another reason for the complexity of the Pseudo Random Code, a reason that's crucial to making GPS economical. The codes make it possible to use "information theory" to "amplify" the GPS signal. And that's why GPS receivers don't need big satellite dishes to receive the GPS signals. We glossed over one point in our silly 'Stairway to Heaven' analogy. It assumes that we can guarantee that both the satellite and the receiver start generating their codes at exactly the same time. But how do we make sure everybody is perfectly synced? Stay tuned and see.

Timing - Achieving Perfect Timing
1. Accurate timing is the key to measuring distance to satellites. 2. Satellites are accurate because they have atomic clocks on board.

3. Receiver clocks don't have to be too accurate because an extra satellite range measurement can remove errors Discussion If measuring the travel time of a radio signal is the key to GPS, then our stop watches had better be darn good, because if their timing is off by just a thousandth of a second, at the speed of light, that translates into almost 200 miles of error! On the satellite side, timing is almost perfect because they have incredibly precise atomic clocks on board. But what about our receivers here on the ground? Remember that both the satellite and the receiver need to be able to precisely synchronize their pseudo-random codes to make the system work. (to review this point click here) If our receivers needed atomic clocks (which cost upwards of $50K to $100K) GPS would be a lame duck technology. Nobody could afford it. Luckily the designers of GPS came up with a brilliant little trick that lets us get by with much less accurate clocks in our receivers. This trick is one of the key elements of GPS and as an added side benefit it means that every GPS receiver is essentially an atomic-accuracy clock. The secret to perfect timing is to make an extra satellite measurement. That's right, if three perfect measurements can locate a point in 3-dimensional space, then four imperfect measurements can do the same thing. Extra Measurement Cures Timing Offset If our receiver's clocks were perfect, then all our satellite ranges would intersect at a single point (which is our position). But with imperfect clocks, a fourth measurement, done as a cross-check, will NOT intersect with the first three. So the receiver's computer says "Uh-oh! there is a discrepancy in my measurements. I must not be perfectly synced with universal time." Since any offset from universal time will affect all of our measurements, the receiver looks for a single correction factor that it can subtract from all its timing measurements that would cause them all to intersect at a single point. That correction brings the receiver's clock back into sync with universal time, and bingo! you've got atomic accuracy time right in the palm of your hand. Once it has that correction it applies to all the rest of its measurements and now we've got precise positioning. One consequence of this principle is that any decent GPS receiver will need to have at least four channels so that it can make the four measurements simultaneously. With the pseudo-random code as a rock solid timing sync pulse, and this extra measurement trick to get us perfectly synced to universal time, we have got everything we need to measure our distance to a satellite in space.

But for the triangulation to work we not only need to know distance, we also need to know exactly where the satellites are. In the next section we'll see how we accomplish that.

Satellite Tracking
    To use the satellites as references for range measurements we need to know exactly where they are. GPS satellites are so high up their orbits are very predictable. Minor variations in their orbits are measured by the Department of Defense. The error information is sent to the satellites, to be transmitted along with the timing signals.

Discussion Thus far we've been assuming that we know where the GPS satellites are so we can use them as reference points. But how do we know exactly where they are? After all they're floating around 11,000 miles up in space. That 11,000 mile altitude is actually a benefit in this case, because something that high is well clear of the atmosphere. And that means it will orbit according to very simple mathematics. The Air Force has injected each GPS satellite into a very precise orbit, according to the GPS master plan. On the ground all GPS receivers have an almanac programmed into their computers that tells them where in the sky each satellite is, moment by moment. The basic orbits are quite exact but just to make things perfect the GPS satellites are constantly monitored by the Department of Defense. They use very precise radar to check each satellite's exact altitude, position and speed. The errors they're checking for are called "ephemeris errors" because they affect the satellite's orbit or "ephemeris." These errors are caused by gravitational pulls from the moon and sun and by the pressure of solar radiation on the satellites. The errors are usually very slight but if you want great accuracy they must be taken into account. Once the Department of Defense has measured a satellite's exact position, they relay that information back up to the satellite itself. The satellite then includes this new corrected position information in the timing signals it's broadcasting.

So a GPS signal is more than just pseudo-random code for timing purposes. It also contains a navigation message with ephemeris information as well. With perfect timing and the satellite's exact position you'd think we'd be ready to make perfect position calculations. But there's trouble afoot. You can't manage what you don't measure - use GPS fleet tracking

Handling Errors
1. The earth's ionosphere and atmosphere cause delays in the GPS signal that translate into position errors. See a summary of error sources. 2. Some errors can be factored out using mathematics and modeling. 3. The configuration of the satellites in the sky can magnify other errors. 4. Differential GPS can eliminate almost all error. Discussion Up to now we've been treating the calculations that go into GPS very abstractly, as if the whole thing were happening in a vacuum. But in the real world there are lots of things that can happen to a GPS signal that will make its life less than mathematically perfect. To get the most out of the system, a good GPS receiver needs to take a wide variety of possible errors into account. Here's what they've got to deal with. First, one of the basic assumptions we've been using is not exactly true. We've been saying that you calculate distance to a satellite by multiplying a signal's travel time by the speed of light. But the speed of light is only constant in a vacuum. As a GPS signal passes through the charged particles of the ionosphere and then through the water vapor in the troposphere it gets slowed down a bit, and this creates the same kind of error as bad clocks. There are a couple of ways to minimize this kind of error. For one thing we can predict what a typical delay might be on a typical day. This is called modeling and it helps but, of course, atmospheric conditions are rarely exactly typical. Another way to get a handle on these atmosphere-induced errors is to compare the relative speeds of two different signals. This "dual frequency" measurement is very sophisticated and is only possible with advanced receivers. Trouble for the GPS signal doesn't end when it gets down to the ground. The signal may bounce off various local obstructions before it gets to our receiver. This is called multi-path error and is similar to the ghosting you might see on a TV. Good receivers use sophisticated signal rejection techniques to minimize this problem. Trouble for the GPS signal doesn't end when it gets down to the ground. The signal may bounce off various local obstructions before it gets to our receiver.

This is called multi-path error and is similar to the ghosting you might see on a TV. Good receivers use sophisticated signal rejection techniques to minimize this problem. Satellite Errors Even though the satellites are very sophisticated they do account for some tiny errors in the system. The atomic clocks they use are very, very precise but they're not perfect. Minute discrepancies can occur, and these translate into travel time measurement errors. And even though the satellites positions are constantly monitored, they can't be watched every second. So slight position or "ephemeris" errors can sneak in between monitoring times. Basic geometry itself can magnify these other errors with a principle called "Geometric Dilution of Precision" or GDOP. It sounds complicated but the principle is quite simple. There are usually more satellites available than a receiver needs to fix a position, so the receiver picks a few and ignores the rest. If it picks satellites that are close together in the sky the intersecting circles that define a position will cross at very shallow angles. That increases the gray area or error margin around a position. If it picks satellites that are widely separated the circles intersect at almost right angles and that minimizes the error region. Good receivers determine which satellites will give the lowest GDOP.

Timekeeping - Bringing precise time to the world
Although GPS is well-known for navigation, tracking, and mapping, it's also used to disseminate precise time, time intervals, and frequency. Time is a powerful commodity, and exact time is more powerful still. Knowing that a group of timed events is perfectly synchronized is often very important. GPS makes the job of synchronizing our clocks easy and reliable. There are three fundamental ways we use time. As a universal marker, time tells us when things happened or when they will. As a way to synchronize people, events, even other types of signals, time helps keep the world on schedule. And as a way to tell how long things last, time provides and accurate, unambiguous sense of duration. GPS satellites carry highly accurate atomic clocks.

And in order for the system to work, our GPS receivers here on the ground synchronize themselves to these clocks. That means that every GPS receiver is, in essence, an atomic accuracy clock. Astronomers, power companies, computer networks, communications systems, banks, and radio and television stations can benefit from this precise timing. One investment banking firm uses GPS to guarantee their transactions are recorded simultaneously at all offices around the world. And a major Pacific Northwest utility company makes sure their power is distributed at just the right time along their thousands of miles of transmission lines. Beagle Software offers the Star Sync product which sets any Windows computer to the GPS clock and can serve as a time source for computers on a network. We hope you've enjoyed our brief discussion about the technology behind GPS.

How GPS Works
The Global Positioning System (GPS) is a technical marvel made possible by a group of satellites in earth orbit that transmit precise signals, allowing GPS receivers to calculate and display accurate location, speed, and time information to the user. By capturing the signals from three or more satellites (among a constellation of 31 satellites available), GPS receivers are able to use the mathematical principle of trilateration to pinpoint your location. With the addition of computing power, and data stored in memory such as road maps, points of interest, topographic information, and much more, GPS receivers are able to convert location, speed, and time information into a useful display format. GPS was originally created by the United States Department of Defense (DOD) as a military application. The system has been active since the early 1980s, but began to become useful to civilians in the late 1990s. Consumer GPS has since become a multi-billion dollar industry with a wide array of products, services, and Internet-based utilities. GPS works accurately in all weather conditions, day or night, around the clock, and around the globe. There is no subscription fee for use of GPS signals. GPS signals may be blocked by dense forest, canyon walls, or skyscrapers, and they don’t penetrate indoor spaces well, so some locations may not permit accurate GPS navigation. GPS receivers are generally accurate within 15 meters, and newer models that use Wide Area Augmentation System (WAAS) signals are accurate within three meters. While the U.S. owned and operated GPS is currently the only active system, five other satellitebased global navigation systems are being developed by individual nations and by multi-nation consortiums.

Global Positioning System (GPS) Technology and Cars
Over the years, the technology involved in manufacturing an automobile has become more advanced, as automakers shift their focus from basic transportation to the design of features that make a vehicle safer, more comfortable, and more easily operated. One such feature is the global positioning system (GPS). A GPS unit consists of a space segment, a control segment, and a user segment. The space segment is a constellation of two dozen satellites orbiting the earth twice every 24 hours, at approximately 10,900 nautical miles above the earth's surface (1). These satellites are funded and controlled by the U.S. Department of Defense. The control segment is a series of monitoring stations located at different sites on earth. These stations update and correct errors in the navigational message of the satellites. The user segment is a receiver that receives radio waves from the satellites in orbit. It can determine how far away it is from each satellite by keeping track of the time it takes for a radio wave to travel from the satellite to the receiver (2). Four satellites are used simultaneously to pinpoint the precise position of the receiver on the earth. Information from the first three satellites narrows down the range of possible locations to two points; one of these is usually illogical and indicates a point not on the earth. A fourth satellite is used to confirm the target location (3). The accuracy of a typical GPS receiver is about 10-15 meters. This may not be practical for locating a small object such as an automobile, which is about three meters long. Differential GPS (DGPS) is a system that improves the accuracy of the GPS receiver to about one to two meters (4). Several reference GPS receivers are placed at stationary locations, whose coordinates are known. These receivers compare their known locations to the location information they receive from satellites, and broadcast the range errors they detect from each other and from every satellite. A DGPS receiver can pick up this range error information and correlate it with the satellite signals it is receiving, to find out its true position (5). The accuracy is dependent on how fast the reference receivers broadcast their signals. When installed in a car, a GPS unit can provide useful information about the car's position and the best travel routes to a given destination by linking itself to a built-in digital map (6). A monitor in the car shows the relevant portion of the map. The driver can enter the target location, and the computer will calculate the optimal route and display it instantly. It can respond to user preferences and map a route that avoids highways or avoids local roads. If the map is detailed enough, it will also provide the locations of the nearest gas station, supermarket, restaurant, hotel, and ATM machine. Some GPS units can issue auditory directions (i.e., "Turn left,") to guide drivers as they travel (7). GPS also tracks the distance traveled on a particular trip, vehicle mileage, and speed. It can keep a record of driving activity, including the address of each destination, names of streets traveled, and how long the vehicle remained at each location, to allow owners to monitor the use of their cars by other drivers (8). Some systems issue warnings when the car is speeding and when the car is approaching a speed trap (9).

GPS can aid in the recovery of a stolen car. Integrated with the automobile security system, GPS can notify the car owner by phone or e-mail when the car alarm is triggered, and indicate the location of the car (10). An owner can contact the GPS unit from outside the car using a cell phone or via the Internet, and can start the car engine remotely and unlock the door, solving the problem of frozen locks. GPS also monitors the car condition, and issues warnings when the battery is low or when it is time for an oil change (11). The Ford automobile company has developed a new telematic system through GPS on the Ford Focus, that will alert emergency services when an airbag deploys. The emergency services can locate the vehicle quickly and provide assistance to an injured driver or passengers (12). In addition to its applications in private vehicles, GPS is being used by commercial shipping companies to speed the delivery of cargo. It allows companies to track their fleets, record the movement of their trucks, and control route planning (13). Car rental companies can equip their cars with GPS technology to find stolen rental cars and charge customers for dangerous conduct as speeding (14). Commercial fishermen can use GPS receivers to keep track of the best spots where they have caught fish. GPS technology is also a valuable aid to targeting locations for military personnel, firefighters, and construction workers (3). The development of GPS technology opens the door to the design of an unmanned vehicle. Equipped with laser and infrared scanners to "see" its environment, such a vehicle could be used for tasks that are considered too time consuming or hazardous for human drivers. Students in the electrical and electronic engineering department of the Nanyang Technological University in Singapore are working on this technology (15). Though GPS technology is gaining popularity in cars, some problems still exist. The biggest problem is the blockage of signal transmission by obstacles such as mountains, high buildings, tunnels, urban canyons, or thick-branched trees. Multipath signals generated by reflections from nearby surfaces or fences can also interfere with the GPS data. Some methods have been introduced to improve these disadvantages. Dead Reckoning (DR) is a way to keep tracking vehicles during periods of blocked transmission. DR combines directional and distance information from a heading sensor in the car and from the car's odometer, and calculates the current position of the vehicle by computing the course steered and speed over the ground from a last known position. DR is used to improve accuracy when GPS transmission is available, in addition to being a backup when transmission is blocked. However, its accuracy tends to drop if the car travels for an extended period without receiving GPS signals (16). GPS technology is advancing very quickly. New models are being developed every year to correct problems in the older versions, focusing on improved accuracy, better reception, and even more user-friendly features.

Triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly (trilateration). The point can then be fixed as the third point of a triangle with one known side and two known angles. Triangulation can also refer to the accurate surveying of systems of very large triangles, called triangulation networks. This followed from the work of Willebrord Snell in 1615–17, who showed how a point could be located from the angles subtended from three known points, but measured at the new unknown point rather than the previously fixed points, a problem called resectioning. Surveying error is minimized if a mesh of triangles at the largest appropriate scale is established first. Points inside the triangles can all then be accurately located with reference to it. Such triangulation methods were used for accurate large-scale land surveying until the rise of global navigation satellite systems in the 1980s. Applications Optical 3d measuring systems use this principle as well in order to determine the spatial dimensions and the geometry of an item. Basically, the configuration consists of two sensors observing the item. One of the sensors is typically a digital camera device, and the other one can also be a camera or a light projector. The projection centers of the sensors and the considered point on the object’s surface define a (spatial) triangle. Within this triangle, the distance between the sensors is the base b and must be known. By determining the angles between the projection rays of the sensors and the basis, the intersection point, and thus the 3d coordinate, is calculated from the triangular relations.

Distance to a point by measuring two fixed angles

Triangulation may be used to calculate the coordinates and distance from the shore to the ship. The observer at A measures the angle α between the shore and the ship, and the observer at B does likewise for β . With the length l or the coordinates of A and B known, then the law of sines can be applied to find the coordinates of the ship at C and the distance d.

The coordinates and distance to a point can be found by calculating the length of one side of a triangle, given measurements of angles and sides of the triangle formed by that point and two other known reference points. The following formulas apply in flat or Euclidean geometry. They become inaccurate if distances become appreciable compared to the curvature of the Earth, but can be replaced with more complicated results derived using spherical trigonometry.

Calculation

Therefore

Using the trigonometric identities tan α = sin α / cos α and sin(α + β) = sin α cos β + cos α sin β, this is equivalent to:

From this, it is easy to determine the distance of the unknown point from either observation point, its north/south and east/west offsets from the observation point, and finally its full coordinates.

History

Liu Hui (c. 263), How to measure the height of a sea island. Illustration from an edition of 1726

Gemma Frisius's 1533 proposal to use triangulation for mapmaking

Nineteenth-century triangulation network for the triangulation of Rhineland-Hesse

Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and gun direction of weapons. The use of triangles to estimate distances goes back to antiquity. In the 6th century BC the Greek philosopher Thales is recorded as using similar triangles to estimate the height of the pyramids by measuring the length of their shadows and that of his own at the same moment, and comparing the ratios to his height (intercept theorem);[1] and to have estimated the distances to ships at sea as seen from a clifftop, by measuring the horizontal distance traversed by the line-ofsight for a known fall, and scaling up to the height of the whole cliff.[2] Such techniques would have been familiar to the ancient Egyptians. Problem 57 of the Rhind papyrus, a thousand years earlier, defines the seqt or seked as the ratio of the run to the rise of a slope, i.e. the reciprocal of gradients as measured today. The slopes and angles were measured using a sighting rod that the Greeks called a dioptra, the forerunner of the Arabic alidade. A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, the Dioptra of Hero of Alexandria (c. 10–70 AD), which survived in Arabic translation; but the knowledge became lost in Europe. In China, Pei Xiu (224–271) identified "measuring right angles and acute angles" as the fifth of his six principles for accurate map-making, necessary to accurately establish distances;[3] while Liu Hui (c. 263) gives a version of the calculation above, for measuring perpendicular distances to inaccessible places.[4][5] In the field, triangulation methods were apparently not used by the Roman specialist land surveyors, the agromensores; but were introduced into medieval Spain through Arabic treatises on the astrolabe, such as that by Ibn al-Saffar (d. 1035).[6] Abu Rayhan Biruni (d. 1048) also introduced triangulation techniques to measure the size of the Earth and the distances between various places.[7] Simplified Roman techniques then seem to have co-existed with more sophisticated techniques used by professional surveyors. But it was rare for such methods to be translated into Latin (a manual on Geometry, the eleventh century Geomatria incerti auctoris is a rare exception), and such techniques appear to have percolated only slowly into the rest of Europe.[6] Increased awareness and use of such techniques in Spain may be attested by the medieval Jacob's staff, used specifically for measuring angles, which dates from about 1300; and the appearance of accurately surveyed coastlines in the Portolan charts, the earliest of which that survives is dated 1296.

Gemma Frisius and triangulation for mapmaking
On land, the Dutch cartographer Gemma Frisius proposed using triangulation to accurately position far-away places for map-making in his 1533 pamphlet Libellus de Locorum describendorum ratione (Booklet concerning a way of describing places), which he bound in as an appendix in a new edition of Peter Apian's best-selling 1524 Cosmographica. This became very influential, and the technique spread across Germany, Austria and the Netherlands. The astronomer Tycho Brahe applied the method in Scandinavia, completing a detailed triangulation in 1579 of the island of Hven, where his observatory was based, with reference to key landmarks on both sides of the Øresund, producing an estate plan of the island in 1584.[8] In England Frisius's method was included in the growing number of books on surveying which appeared from the middle of the century onwards, including William Cunningham's Cosmographical Glasse (1559), Valentine Leigh's Treatise of Measuring All Kinds of Lands (1562), William Bourne's Rules of Navigation (1571), Thomas Digges's Geometrical Practise named Pantometria (1571), and John Norden's Surveyor's Dialogue (1607). It has been suggested that Christopher Saxton may have used rough-and-ready triangulation to place features in his county maps of the 1570s; but others suppose that, having obtained rough bearings to features from key vantage points, he may have estimated the distances to them simply by guesswork.[9]

Willebrord Snell and modern triangulation networks
The modern systematic use of triangulation networks stems from the work of the Dutch mathematician Willebrord Snell, who in 1615 surveyed the distance from Alkmaar to Bergen op Zoom, approximately 70 miles (110 kilometres), using a chain of quadrangles containing 33 triangles in all. The two towns were separated by one degree on the meridian, so from his measurement he was able to calculate a value for the circumference of the earth – a feat celebrated in the title of his book Eratosthenes Batavus (The Dutch Eratosthenes), published in 1617. Snell calculated how the planar formulae could be corrected to allow for the curvature of the earth. He also showed how to resection, or calculate, the position of a point inside a triangle using the angles cast between the vertices at the unknown point. These could be measured much more accurately than bearings of the vertices, which depended on a compass. This established the key idea of surveying a large-scale primary network of control points first, and then locating secondary subsidiary points later, within that primary network. Snell's methods were taken up by Jean Picard who in 1669–70 surveyed one degree of latitude along the Paris Meridian using a chain of thirteen triangles stretching north from Paris to the clocktower of Sourdon, near Amiens. Thanks to improvements in instruments and accuracy, Picard's is rated as the first reasonably accurate measurement of the radius of the earth. Over the next century this work was extended most notably by the Cassini family: between 1683 and 1718 Jean-Dominique Cassini and his son Jacques Cassini surveyed the whole of the Paris meridian from Dunkirk to Perpignan; and between 1733 and 1740 Jacques and his son César Cassini undertook the first triangulation of the whole country, including a re-surveying of the meridian arc, leading to the publication in 1745 of the first map of France constructed on rigorous principles.

Triangulation methods were by now well established for local mapmaking, but it was only towards the end of the 18th century that other countries began to establish detailed triangulation network surveys to map whole countries. The Principal Triangulation of Great Britain was begun by the Ordnance Survey in 1783, though not completed until 1853; and the Great Trigonometric Survey of India, which ultimately named and mapped Mount Everest and the other Himalayan peaks, was begun in 1801. For the Napoleonic French state, the French triangulation was extended by Jean Joseph Tranchot into the German Rhineland from 1801, subsequently completed after 1815 by the Prussian general Karl von Müffling. Meanwhile, the famous mathematician Carl Friedrich Gauss was entrusted from 1821 to 1825 with the triangulation of the kingdom of Hanover, for which he developed the method of least squares to find the best fit solution for problems of large systems of simultaneous equations given more real-world measurements than unknowns. Today, large-scale triangulation networks for positioning have largely been superseded by the Global navigation satellite systems established since the 1980s. But many of the control points for the earlier surveys still survive as valued historical features in the landscape, such as the concrete triangulation pillars set up for retriangulation of Great Britain (1936–1962), or the triangulation points set up for the Struve Geodetic Arc (1816–1855), now scheduled as a UNESCO World Heritage Site.

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