 In this video we're going to talk about a few other ways that we can represent places in three-dimensional space. We've already talked about a Cartesian coordinate system, which is a system of three perpendicular axes that we use to represent three-dimensional space. But now we're going to talk about another set or a family of bases, polar, cylindrical, and spherical coordinates that can be used to also represent places in three-dimensional space, but in a way that can be more intuitive for certain types of uses, particularly uses in something like aerospace engineering where we're trying to figure out which direction to move in, etc. So once again we are going to define our basis using an origin, setting a place in space as a starting reference, a reference origin, a reference orientation, some direction that we all refer to, and then we'll set some scales that measure how we measure in various directions. So we're going to start in two dimensions with a system called a polar coordinate system. So again this is just in two dimensions and to start our polar coordinate system let's go and look at our basis, what we need. First of all we need an origin. So we start the origin with something called a pole, polar coordinate system. There's our pole, a place in space. Next we need an orientation, a direction that we're going to associate with that pole. So I'm going to go ahead and make a direction. Now there's actually, here's what's difficult about this. There are two systems that are typically in use a lot. One of them you learn in a math class and one of them you learn in a geography class or we use in mapping in other contexts. So I'm going to call one the math system and one the geography system. So we're going to start here with the math system. I'll put the math system on this side and I'll put the geography system on this side. And it's a little unfortunate but these two systems have different conventions on how we typically represent things. So you have to be very clear about which basis you have chosen to use even if they are both polar bases. So I'm going to start with the math system and typically in the math system we extend from the pole an arrow off to the right. This is called a ray and this is our polar direction. This is a direction associated with the pole. It starts to notice it only goes in one direction unlike the Cartesian coordinate systems where we go positive and negative in both directions. The polar direction only goes in one direction and we'll call that our positive direction for our pole. For our geography system we do almost the same thing. We typically start with our pole but now we typically go in a direction that's associated with north and if we're drawing on a screen like we are here north is often indicated as up. So usually the two systems already differ in what our directions are, our orientation. Notice there's an orientation that's also associated with the plane that we're in. In this case my plane is the screen in front of me. If I had a piece of paper that plane would be the plane of the paper and maybe that's laying flat on the ground but there's an association that's already sort of implicit in that we're in a two-dimensional system and that system is flat. Now we're not quite done yet because remember this is a two-dimensional system, two dimensions meaning we need two sort of ways of measuring and really we've only pointed in one direction. So once we've established our orientation as far as an arrow in one direction we are going to in the same way as in the Cartesian system we are going to move in one direction along a certain scale. In fact let's go ahead and move on to the scale for that. I can establish a scale of a unit of a distance in that direction, one unit in that direction and usually we'll label this with zero and one, zero and one there is a single unit in that direction. So there is a length associated, there's one length associated with our two-dimensional polar system, one length. But what makes the system different from our Cartesian system is that the other measure we use is not a length, instead it's an angle. What we are going to do is try to quantify an amount of rotation. In the case of our math system when you first learn angles you typically rotate up, you take this dimension here and you rotate up a certain amount and that is an angle measurement. So we have one length and one angle. In the case of our math system that angle is typically counter-clockwise opposite a typical analog clock. So we will rotate in this direction. In the case of our geography system it's the opposite direction. We start north and then we tend to rotate in a clockwise direction and measure our angles down in a clockwise direction. This creates a system of two measures. Our scale for that is measures of angles and we have two typical measures of angles, either in degrees where all the way around represents 360 degrees or in radians where all the way around represents the number 2 pi. So for our two-dimensional system we have a length, typically that length is represented with the letter r. Why r you say? Well r we can think about this is if we go all the way around this angle we create a circle and so r is considered the radius the distance to that circle. So r is often used to represent the radius and so we'll use that to represent our first measure and that's a scaled measurement of length. And then we will also have our angle and we'll typically use a letter. One of them that's used very often is the Greek letter theta. So there is our polar system, two dimensions and two measurements r and theta. So now let's move from a two-dimensional system to a three-dimensional system. To do so I'm going to take my piece of paper here and again set it up showing that we have our polar direction and we have this plane, this two-dimensional plane that we've already associated with this. This is actually called our reference plane. We will need one of these as part of our orientation and we will start by putting in origin a pole onto this reference plane. I'm going to take that plane and I'm going to flatten it out and lay it down horizontally. Now from there I have two ways of thinking about how I want to take care of that third dimension, this vertical dimension here. There are two ways of dealing with it and the first one creates what we call the cylindrical coordinate system. In the cylindrical coordinate system we assume that we have this circle here that we just talked about where we rotate around at some point at the circle. We go out to the place in the circle, we rotate around somewhere and then we have to get into this third dimension and it's relatively simple in cylindrical coordinate systems because what we do then is we simply go straight up. We go vertically up so I'm going to try to redraw this system here just for a second. We'll say we have, there's my polar direction and I'll kind of lay out this overall shape here of my piece of paper drawing it at a bit of an angle just like we did in the Cartesian coordinate system. Then if we assume that we come perpendicularly straight up off of that piece of paper, straight up perpendicular, we make that perpendicular. Now we have the same measurements that we used in polar, our radius, our angle, so this is a length and this is an angle, but then our third measurement is we create a new scale up something along those lines and we measure that distance up off of that flat plane. We come up off the flat plane. So if I want to find a place in space I go out a certain distance, I rotate along my angle and then I come up to a certain place and that third one, that third height will represent with z just like x, y, and z in our Cartesian coordinate system will typically represent that with z, that is another length. So to find something in this three-dimensional space in a cylindrical coordinate system, again cylindrical is 3D whereas polar is only 2D, we'll need three pieces of information. Some form of rotation in our reference plane, a distance along our reference plane, this radius, and then a distance up and those three pieces create a three-dimensional basis known as the cylindrical coordinate system. Now we'll move on to our third basis, our spherical coordinate system. This is a three-dimensional system as well, so we will require three pieces of information. In both the math and the geography contexts, we again start from our reference plane. We do have a reference plane, but it depends on which direction you sort of want to orient, again just like in the polar coordinate system there are different conventions and there's also different conventions on how this is done in three dimensions. Let's start with the math system, although in something like aerospace engineering we will typically use the geography system more frequently. In the math system, we not only have our reference plane, so in both these cases we have a reference plane and a reference direction, but we will start with a, just as before in the cylindrical system, we will start with the vertical axis. But instead of this axis being an axis we're going to measure up and along, this is going to be an axis that we're going to use as a reference for rotation. This particular axis is typically perpendicular to our reference plane, okay, and this is called our zenith axis. So the way we work in the math system is typically what we do is we start from that zenith axis which we think about as being up, and then one of our first measures is an angle called the zenith angle or the inclination angle. And what we do is we come down off of that angle, so we start vertically and we come down off of that angle toward the horizontal. If we keep going past the horizontal, we can keep going all the way till we're pointing down, so this angle can range from zero degrees pointing straight up to 180 degrees pointing straight down. So that zenith angle or that inclination angle is the first of our angles for the spherical coordinate system. Now this differs from the geography system. In the case of the geography system we are actually going to assume that we're in the plane, that we start horizontal on the plane, and then we come up or down above or below the horizontal. So in the case of our geography system we're going to be coming up, and whereas this is called an inclination angle, our geography system, this is called an elevation angle, where we're looking at the horizon and we come up off the horizon to measure an elevation. So that's very different ways of thinking about things and it's very important. Notice there is a relationship between these two things, the combination of the elevation angle and the inclination angle. If they come together to the same direction, then they're going to add up to 90 degrees, but those are obviously very different angles, so you need to know which system you're working in. So once we have that inclination angle what we also do then is as we come down to that space, then what we're going to do is we're going to come down to that space above our longitudinal polar direction, then we're going to rotate from that polar direction a certain amount just like in our polar coordinate system, and then we're going to go out a certain distance. So we've come down, we've rotated, and then we're going to end up in a certain position. Now again it's hard to do this in 3D, so let's think about it from this perspective. I come down, I rotate, and then from there I'm going to go from a point at my origin out to the location, whatever that location is I want, how far out I want to go, I can keep going, or I can stop somewhere along this particular line. Okay, and that is our distance, and typically we represent that distance with the Greek letter rho, kind of like the r for radius, but we want to distinguish it from the two-dimensional distance, so we actually call it r, we use the Greek letter rho to represent that particular thing. Okay, and so then we write these three angles, the three pieces together, notice we now have, I'm sorry, we don't write these three angles, we write this one length, rho is a length, and two angles. Cartesian system we had three lengths, in our cylindrical system we had two lengths and one angle, and now in our spherical 3D system we have one length and two angles. So in the case of our math system we will often write these with the Greek letter rho to represent that length, the Greek letter phi to represent that inclination angle, and then the Greek letter theta to represent the other, the rotation angle here. For our geography system we are also going to have a similar set, one length and two angles, but we've already set our first angle, we have our elevation angle, we also have that angle of rotation in the plane, but notice that rotation is actually from north and in our clockwise direction, so I'm going to measure that along the plane. So typically the way we write that is we will start with that angle often in geography, and that angle is called the azimuth, and we will still often represent it with the Greek letter theta, but we have the name for it in the geography convention, it's called the azimuth, so we always start by pointing north, that finds our azimuth, then from there we will measure that elevation up off of the plane, there's our elevation and again we will use the Greek letter phi, and then last but not least we will go out a certain distance, once we've raised up we'll come out a certain distance, and there's the location of our point, azimuth, elevation, and then our distance again typically represented with the Greek letter rho, and that's again to distinguish it from the r that we would use in a cylindrical system, which would be in that reference plane. So because we're typically going to use the geography system more frequently, I'd like to rehearse this a little bit with you, what I'd like you to do is where you are, see if you can figure out where north is, make a good guess, and then take your arm and try to point it horizontally toward the north, and I'm going to suggest you use your right arm to do this, so I'm going to turn around so you can follow along with me, I'm going to take my arm and point it, use the little pointer finger here, and point it toward the north. Now if I want to do my azimuth, if I have an azimuth that's going to be my first angle, and that's going to be sliding side to side along that horizontal plane, if I want to go in a positive direction I go toward my right, if I want to go in a negative direction I start at north again and go toward my left. So let's say I wanted to go with an azimuth of say about 70 degrees, that would turn me toward my right, not quite all the way to the right, and I'd be pointing off still a little ahead of me, but toward the right. That would be my first measurement, then I might say okay I would like an elevation, an elevation is going to raise off of the horizontal, if it's a negative value we'll go down, but we're going to use a positive value, let's go up say 40 degrees, so that's going to come up at an angle, and now I'm pointing in a general direction, so I'm going to use my azimuth, my elevation, and now I figure out how far away the thing is from my origin, which in this case is going to be my shoulder, and I'll follow it along, maybe it's only two feet away, or maybe it's 10 miles away, or maybe it's 100 light years away. One of the reasons for using this system is you may not be able to get out there to actually measure that final distance, so we only have one length, and we can do some calculations for that one length, as opposed to having multiple lengths if we're using our cartesian system, or even our cylindrical system. So again, azimuth, elevation, distance, and that is the geography convention for the spherical coordinate system.