Suppose it's your job to relay information about position. Your success depends on being able to convey direction and distance accurately and efficiently. The world, though, is much more complex than 'down the road about a mile and turn left' or pointing 'that way, 2 miles.' We will explore different ways to convey this information, and the situations which lend preference to one system over another.
Think a minute about giving directions. Usually, we have no real problem when we say "front, back, left, right" for directions. However, we must always keep in mind that these are relative directions. For example, your left is my right when we face each other. We usually correct for these differences, but when we don't, the effect can be exactly opposite of what we meant.
For example, it's graduation night, and your parents are having a BBQ for your friends. You e-mail these directions to your friends:
Drive down Broad Street to Alps. Turn left at Alps. Drive half mile down Alps. Turn right at the church. Drive three blocks. Turn right. Third house on left.
Most of your friends make it to your house. However, one friend from your math class winds up lost. He followed your directions to the letter, but began by driving "up" Broad Street... he was coming from the "wrong" direction. You all have a hearty laugh over the misunderstanding, but it does make you think about how something so simple as giving directions can become complicated. As your friends reminisce about the years gone by and project their invincibility and remarkable talents to the future, you are troubled by the mundanities of direction giving. After all, what is giving directions? How can you make it more precise and unambiguous?
Later that night, you sit down and really start thinking about these questions. First, you recognize that the problem is essentially that you begin with two different locations and must try to describe a path between them. The path must account for various parameters... obstructions, traffic patterns, etc. You want it to be a short as practical, easy to follow, and to always end at the destination, no matter where the starting location is. Between any two positions, you then have two parameters: distance and direction. You notice that distance is not too problematic. A mile is a mile. Thinking on this, you realize that the question becomes one of how to describe a position and direction. How you choose to describe a position will dictate how the direction will be described.
Let's look again at your directions and try to see what the issue is:
Drive down Broad Street to Alps. Turn left at Alps. Drive half mile down Alps. Turn right at the church. Drive three blocks. Turn right. Third house on left.
Highlighted in green are distance references, which aren't very problematic. For example, the "third house" might be 300 yards or 2 miles down the road, but it is still the third house. The blue highlights are reference positions. Though the church may be more specifically described (named, for example), these reference locations are real, absolute entities. Again, this is not the source of the misunderstandings.
Now think about the locations in your directions:
Broad Street --> Alps --> church --> 3rd block --> 3rd house. Though some of these are vague, they are still definite, absolute locations.
The pesky elements are those arrows in between the locations. Look at all those red words in the directions! Every one is a relative description. They depend strictly upon the orientation of a person at that position. For example, two people at the exact same location, but facing each other, will use the opposite direction words (whose left? whose forward?) to describe any of the positions in their environment. These work just fine if your visitor maintains the orientation in reference to which you gave the directions. But what about everyone else? Do you really have to give 20 sets of directions in order to have 20 people show up from 20 different places? And if someone stops for gas...
As a brand new member of the working class, you have found a job as a dispatcher for a medical service. Your job is to tell the ambulances where to go. You work in a city that, by an amazing feat of civil engineering, is laid out on a grid. Though everyone else in the city may call this road "Main St." and that road "Broad St," you and the other public service personnel refer to them by numbers and directions. Town Hall is the center of town, and the reference point for the rest of the town. For every location to which you dispatch, you radio four bits of information:
For example, you are stationed at the hospital at (S2, W2). City Hall is at (N0,E0).
Essentially, we have simply applied a familiar Cartesian coördinate system as a method for describing position. Like any Cartesian plane, we have an origin (City Hall), two perpendicular axes, and a grid formed by marking evenly spaced units along each axis. On our map, these units are city blocks.
This system has some basic benefits. It's very easy to use and to get around town. Driving distances are easily computed: for two addresses, you compare each of the coördinates and take the distance between them. When the direction part (N/S, E/W) of the coördinate is the same, the numbers subtract. When they are different, the numbers add. For example, from your station to (N4, W3) is (4+2) + (3-2) = 6 + 1 = 7 blocks. When you radio coördinates to your medics, they know almost automatically to drive x blocks this way, turn left, drive y blocks, and they can easily do this from wherever they happen to be, even if they don't start from the hospital. Given a location's position, anyone can find that position from wherever they may be.
All is not perfect with this system, however. You start thinking about the limitations of the system the first time you have to dispatch a helicopter from your hospital to a potential heart-attack patient across town, at (N7, W7). You're smarter than to tell the pilot to fly north 9 blocks and west 5 blocks. Native to our understanding of the world is the notion that a straight line is the shortest distance between two points, but your map and system doesn't tell you how to find that straight line if your patient is not on the same road as you are. You can figure out how to tell the length of the straight line from you to the patient by using Pythagoras' Theorem, but you surely can't radio the instructions "10.3 blocks, that way" while you point out the window!
Luckily, the pilot is very good at navigation and finds the location without a problem, but you lose sleep thinking about this problem. You are also troubled by the fact that the helicopter only had to fly about 18 and a half blocks, where an ambulance would have driven 26 blocks! You know that they have to stay on streets, but you also realize that while the coördinate information can be used to find distances between objects, it requires a bit more work, and so mistakes are always a factor. You also notice that if your medics don't know their exact location already, they can't tell how to get to the location. Because the directions conveyed through the coördinate system are absolute, measured against the two axes, if a medic is unaware of his or her position relative to both of those axes, he or she will have no means of orienting to the desired location. So you drift to sleep, trying to imagine some way to solve these problems.
After several years learning the ropes of medical dispatch, you decide to try your hand at Search and Rescue (SAR), to which end you find yourself as the SAR coördinator aboard a Coast Guard cutter stationed at the beautiful but treacherous Air Station Astoria, Oregon, where the Columbia River roars into the Pacific Ocean. Your job is the same: convey position by direction and distance. However, the way in which you specify locations has changed. You are no longer in a fixed location, but aboard a moving ship. You have no grid, no roads, no signs from which to reference directions. Since both air and water craft can travel in straight lines between any two points at sea, it is really more useful to give one direction and a distance. No, I'm not suggesting you lean out of the bridge, point, and tell the pilot to go 15 miles "that" way. Essentially, though, that is what you do, but with a bit more sophistication. This system, described below, is known as the polar coördinate system, and offers some advantages to the rectangular system already presented.
Consider a search and rescue scenario. First, an alert is raised: a distress signal comes in from the victim, an observer relays an accident report, or a family member reports someone has not returned from the sea. Next, a search is initiated. This part typically involves aircraft and ships patrolling in a grid pattern to cover large expanses of ocean. Hopefully, contact is made with the lost party, and, if possible, survival gear including a signal beacon is dropped. Finally, the rescue craft is dispatched to the scene for rescue and recovery.
This dispatch is your responsibility, and how you do it is slightly different from how you dispatched an ambulance. Since the rescue craft (air or sea) will travel straight (usually) to the scene, you want them to embark immediately upon dispatch along the correct direction for the fastest response. In other words, you'll never have the craft proceed north, then west, to get to a position north-west of dispatch, as you did with the ground based ambulances. You'll just tell them to go north-west, instead.
Let's imagine the simplest rescue scenario:
A small fishing vessel has lost power and is floundering in choppy seas. The boat is equipped with a radio and a rescue beacon, so the captain radios a distress call and activates the rescue beacon. Electronic equipment aboard your cutter detects the beacon, calculates its position and you dispatch the rescue craft from your cutter to retrieve a profusely-swearing and ego-deflated captain. Because the rescue craft began its dispatch from your location, the pilot needed only to note the direction and distance of the beacon from the cutter and then to proceed in that direction until the rescue scene was reached. This scenario, then, only requires two pieces of information: in what direction is the rescue, and how far away is it? note: even the distance can be left out... the pilot of the rescue craft could just travel along a straight line until it reaches the scene, as long as the straight line is perfectly directed.
A complication: your dispatched craft arrives at the scene, but finds that it cannot rescue all of the crew members of the floundered boat. Now you must dispatch a second craft, but one from a location other than your cutter. The two pieces of information (which direction, how far) that dispatched your craft will not suffice for the second craft, because that information is relative to your own location, and will always be different for any other location.
To overcome this difficulty, we need to think again about the basics of directions. First, notice that because your dispatch system is relative to you and your location, you cannot give precise and accurate directions even to yourself if you are moving. In some ways, this problem is similar to the directions you gave for your graduation party. What you need is an absolute cardinality, something that, no matter where you are, you can always point to it. Thinking in the small scale, you realize it is already done with a boat: everything on the boat is given a position relative to the bow. When you face the bow, you hold out your left arm, and it points to port. When you turn around and face the stern, your left arm now points to starboard.
Walk aboard any ship, no matter what size or class, and ask the sailors to immediately stop whatever they are doing, stand up, and, facing whatever direction they happen to be in, and point to their right. Every one of them is likely to point in a different direction. Repeat the experiment, but now ask them to point to starboard. Everyone points in the same direction this time! This is because every direction on a boat is referenced from a cardinal point, the bow. Everyone knows where the bow is, and everyone is oriented to that reference point.
What works on a ship should work for a sea. Pick a single, unmovable point, that everyone in your fleet knows how to find. Anytime you point at that object, no matter where you are, that is 0°. Everything else is measured from there. For example, when you point at your floundering ship, also point at your landmark. The angle your arms make is called the bearing of the rescue. As a convention, this angle is measured clockwise from the reference point, also called the cardinal point.
Returning to our dispatch situation, we now have specific numbers that tell us where everything else is in relation to the cutter and the reference point. The cardinal point is 0° and everything else is measured by the angle (measured clockwise) made when pointing to both the cardinal point and the desired location. This angle is called the bearing for the location.
All this is well and good aboard your cutter, and no confusion arises for dispatches from your location. Any other location, however, and the situation is a bit more complex. Look at the following map:
It's pretty clear that the rescue craft has a different bearing for the rescue site than you do. This is because of the relative nature of the system: every bearing is measured relative to your particular orientation to the fixed cardinal point. In making bearings easy for one craft, you've traded universal coördinates for relative ones, and you must now "translate" to other frames of reference in order to convey direction. Luckily, though, this translation is not all that hard, involving just a little trigonometry.
All we've done in these two mapping and direction scenarios is to illustrate the differences between rectangular and polar coördinate systems. What should be clear about the two situations is that the two systems aren't weird and abstract concepts. Rather, they both have real-world applications, and each has significant characteristics which lend to being appropriate for a variety of situations.
We have also touched upon one of the greatest and most fruitful tensions within mathematics, the beautiful intertwining of number and space. To the Greeks, these concepts were very disparate. After all, what has a number to do with geometry? Sure, you can measure things, but the Greeks never really thought of that as mathematics. The reason why is that numbers allowed for infinity (and the Greeks felt that could not be with regards to space), while geometry admitted irrational numbers. It was not until Descartes in the 17th century that the two were unified, greatly enriching the study of both.
Too often, we shudder when we are confronted with the polar system, especially since it is usually taught after the rectangular system, and so it seems to be more difficult. However, the polar system is actually more strongly rooted in our native intelligence, our intuitive understanding of the world and our place within it. Ask a child to tell you where something is, and the child will 'point' you to it, a very 'polar' way of conveying direction. Directions containing cardinal terms (left/right or north/south, for example) are of a more rectangular nature, and convey not only a knowledge of direction from the initial location, but also a knowledge of the specific path and geography between the two locations.
When we're first exposed to algebra, and receive our first taste of functions and graphing, we are taught the rectangular system by default. The reason is for simplicity. We're taught about the linear functions, y = mx + b, and so it makes sense in this setting to introduce graphing as plotting (x, y) for values of x. Before long, quadratic elements like parabolas and hyperbolas, and then more general polynomials are introduced, and their graphs are again explored in a rectangular setting. Later, we continue to use the rectangular system as our default for exploring trigonometry and calculus, and we often forget that there are other ways of presenting the information. We become more general, and start plotting (x,f(x)). In many ways, we become so mathematically sophisticated that we forget the simplicity of pointing.
When we explore the polar system, we have to realize that we are changing the presentation of our results, not the results themselves. The miscellany of mathematical functions, trig, polynomials, logarithms, remains unchanged by our choice of graphing methods. When we choose to use the polar system, though, we choose not to display the relation between x and y (or x and f(x)). Instead, we graph (
As we explore mathematical applications, particularly in physics, we are often bombarded, it seems, with the polar system. Many times, it is almost as if we have to relearn all the math we've learned up to that point, and it sometimes doesn't seem very obvious why it is worth the trouble. However, if we think about how we observe our physical world, we see that it really is of a polar nature, and the mathematics is then native to the polar system. Extrapolating from our naïve child pointing a finger, we can think about a scientist making any observation. Whether it's with laser, radar, sonar, or electron microscope, the physicist directs a straight-line beam (of light, radio waves, sound, or electrons) at an object, and observes the reflection of that beam as it returns. By knowing the speed of that beam and measuring the elapsed time between emission, reflection, and detection, the scientist then knows how far away the object is. Choosing some convenient cardinal point for reference, the scientist is natively at the origin of a polar system. The observed data, then, is already in polar form: a beam of a specifically known length pointing in some direction. It makes perfect sense that we would want to be able to manipulate the data in this native system, without recourse to converting it to rectangular coördinates.
Often, it is the case in mathematics that the rigorous description of a concept is developed much later than the intuitive and practical sense of the subject. Calculus, for example, was developed in the 17th century, but had to wait until the 19th century for a rigorous basis, the study of which we now call analysis. This is particularly true of the polar coödinate system. Any child knows how to use it, and demonstrates this every time he or she points at something to locate it, but many adults are ignorant of its applications and advantages.
Advantages?! In the scenario concerning the three crafts, we had to manipulate a lot of trigonometry simply to relay information about a location relative to different vessels, which certainly doesn't seem like an advantage. While it is easy to see that the polar system has an intuitive basis, we struggle to see the computational advantage of using it. When this concept is developed, we find that the universe is now prime for exploration, as all we can do as scientist is observe and calculate, point our gadgets, and measure.
Enter 18th century Swiss born mathematician Leonard Euler (pronounced "oiler"), one of the greatest minds in the field. Among the many contributions was an elegant unification of the polar system, complex numbers, and what we now call Euler's constant, e. Though too far afield to explore, we can still appreciate the rudiments of the connection: eiθ = cos θ + isin θ. We have θ as our desired bearing angle, and while i is more commonly used to denote the imaginary axis, we can also think of it as the perpendicular of our cardinal line. This is often called the cis function, and stands as one of the most useful tools a physicist has to relate to the world. The cis function makes possible easy manipulations, such as calculus and trigonometry, in the native polar system of physical data. After learning this method, the polar system becomes more intuitive and calculations are much more easily accomplished without resorting to conversion to rectangular systems. This allows for the development of an entire brach of mathematics, vector analysis, which is based on the notion of remaining stationary at an origin, and performing calculations by pointing and noting distances and angles. This seemingly complicated form of math, tied to the dreaded polar system, is no more than making rigorous a child standing in the middle of a room and telling where everything is.
Other career usages for polar systems here.
A further exploration of complex numbers here.