 Hi, I'm Zor. Welcome to a new Zor education. We are going to talk today about how we position ourself on a sphere using some numerical coordinates in somewhat equivalent, like we have for instance, Cartesian coordinates or polar coordinates on the plane. So we have to identify our place, our location on a sphere. Now, obviously, the most important application of this is navigation on our planet, on Earth. So, first of all, let me just make a couple of assumptions. Assumption number one, I'm simplifying our planet to having a shape of ideal sphere. Obviously, it's not and therefore for real navigation, whatever I'm talking about should be slightly modified. But that's not really subject of this lecture. Subject of this lecture is to have a sphere and to have coordinates on that sphere. I'm talking about mathematically ideal sphere. However, certain resemblance to coordinates on Earth, longitude and latitude, obviously, will be used and that's why it's very important to basically consider whatever I'm talking right now about as a good model, a theoretical foundation for real application, which is actually navigation on Earth. All right, so my first assumption is we are talking about ideal sphere. Now, the second assumption is that in the game, it borrowed from the real life from our planet. I'm assuming that there is something which we can call an axis because, you know, in sphere we can have any diameter, basically, it can be an axis. Well, in this particular case, we assume that there is one specific diameter of a sphere which is an axis and I will call it an axis of rotation or whatever. It doesn't really matter. What does matter is that I will call the two points where this diameter intersects the surface of a sphere, north pole and south pole. So one particular diameter in the sphere is chosen. I call it an axis of rotation and two opposite points are called north and south. Doesn't matter which one is which because we're talking about sphere, right? I present it on this particular picture in more or less the same fashion as it is presented in geography north and the top and south and the bottom, but again, doesn't really matter from the mathematical standpoint. So now on this sphere, I would like to introduce certain system of coordinates. Now, how I'm going to do it. Okay, so number one is I will introduce a concept of meridian. So here is the point on the surface of a sphere and I'm talking about meridian which is going through this sphere. Now, how to determine the meridian? I will take a half plane which goes through this line which is basically the boundary of this half plane and this point. So the line and the point always define one and only one unique plane and half of this plane which is divided by this line where my point is located is the half plane I'm considering right now. Now this half plane intersects my sphere along some curve. Now, what is an intersection of a plane and the sphere? Well, actually we can talk about this as a problem or something. It's very easy to prove that this is a circle. So this is half a circle. The other half a circle will be with another half plane which we don't really consider. Now, this intersection of the plane which, half a plane, which goes through the axis and the point we have chosen this intersection is called a meridian. Okay. Now, obviously each point has a meridian which goes through this point. There might be a situation when different points are in the same meridian or a different meridian. Now, now the question is how can I identify the meridian? If I will identify the meridian, I will reduce the location from the entire sphere to only one particular curve on that sphere. Then I will need another coordinate to define within the curve but right now let me first define the curve. Define the meridian where our point is located. How can I do it? Well, here is a very simple way of doing it. Let's fix one particular meridian which goes through one particular point. Now, this is my fixed meridian. It means that I have fixed a particular half plane which goes through some point. Now, in practical situation, the reason meridian which is really fixed and it's called Greenwich meridian, it goes through Royal Astronomical Laboratory, I think it's called in Greenwich, which is right now part of London. So one particular meridian is chosen, which means that the half plane is chosen. Now, if I have a location one half plane, how can I measure or identify rather the location of another half plane which also goes through the same axis? So its situation is like this. This is my axis. This is one half plane, okay, and this is let's say another half plane. Now, how can I identify one against another? Oops, that's a wrong one. One should be from here. Okay, so these are two half planes and obviously they form dihedral angle which we can measure. Now, the way how it's measured is the following. From this fixed meridian, Greenwich meridian in practical in geography. Everything which goes towards east and again I'm talking about geographical east and west. So if you go and if you look from the north pole, it will look like counter clockwise, all right. So all these angles will be called east from the Greenwich meridian and all the half planes which are positioned to the west of Greenwich meridian will be called west obviously. And I can measure an angle, a dihedral angle from the Greenwich meridian into one direction up to 180 degree to an opposite and another, so from the minus 180 degree to plus 180 degree, I can measure this dihedral angle. So this one is called west, right? So west and this one is called 100 degree east. So by specifying a dihedral angle of this half plane relative to the half plane defined by Greenwich meridian or zero meridian, whatever you want to call it. By identifying this angle, I'm actually defining exactly the meridian which goes through the point which I'm interested in. So from this Greenwich meridian I go either east up to 180 degree or west up to 180 degree. So that's the definition of my meridian and this particular dihedral angle, whatever the dihedral angle is, is called longitude. All right, so longitude is a measure in usually in degrees with a specification of direction of the half plane which defines meridian going through my point of interest relative to meridian, which is a zero meridian, Greenwich meridian. All right, so that defines my meridian. Now I have to define the point on the meridian I have to identify and for this we use the second dimension. Second dimension is as follows. Now if I will have a plane which goes perpendicular to the axis through the center, this one. Now this plane cuts it horizontally so to speak according to my picture when north and the top and south is on the bottom. Now the intersection of this plane with a sphere is called equator. Now equator intersects obviously every meridian and it defines a zero point on that meridian. Now everything should be either up or below this zero point. So the way how I will identify the point on this meridian is the following. I will draw planes parallel to the equatorial planes. Everywhere. Position of this plane actually defines position of my point on the meridian, right? So if I draw the plane through the point of my interest perpendicularly to the axis it will be some kind of a circle and again I have to identify this plane which produces this particular circle which goes through this horizontal plane perpendicular to the axis. That plane would identify this point. How can I identify the position of the of the plane? Well, there are many different ways, but what traditionally is used is an angle from the zero to this point to any point on this plane including this one of this. Now the angle relative to the equatorial plane. So if I will draw a radius from the from the center of a sphere to the point of interest. Now this is the straight line, right? Now it forms an angle with the equatorial plane because this is the center and the center is connected to this point and also the center is part of the equatorial plane. So we have a situation where we have a plane, equatorial plane and some kind of a straight line from this and I'm talking about an angle between this particular line, which means you have to draw a projection onto the plane and this is an angle. Now obviously if this point is on the equator itself, which means within the equatorial plane, the angle is zero. Now if my point is on the north pole, then this is perpendicular to the equatorial plane, right? So it will be 90. Which means that basically the angle, this angle is changing from 90 degrees south, which is this one, towards zero and towards 90 degrees north. With equator having this, and by the way, it's called latitude, with equator having latitude of zero degrees. Poles have latitude of 90 degrees north and south and everything in between has about everything in between. So right now I'm talking about identifying the position of my point with two things. The meridian, which goes through this point, position of which we can identify relative to the Greenwich meridian. And the parallel. Now the intersection of this plane I was talking about with a sphere is called parallel. So the parallel is an intersection of this plane with the surface of the sphere. So the intersection of parallel and meridian gives our point. Position of the parallel is obviously defined by this angle of any point on this parallel, including ours with equatorial plane. So these are two coordinates, which I was talking about. The longitude, position of the meridian which goes through the point and latitude, position of the parallel which goes through this point. Now, what else is interesting? Now I was talking about measuring in degrees. Now, obviously, you know that every degree can be divided into minutes and minutes into seconds. So we have one degree is equal to 60 minutes and one minute is equal to 60 seconds. That's how it's measured. Now, how big is the second? Well, I'm not talking about time. I'm talking about geography right now. Well, let's consider that equator has some lengths. So if we divide this length into 360 degrees, you will have the distance from one degree to another degree on the equator. Now as far as I remember equator is about 40,000 kilometers. So if you divide it by 360, it's about 111 kilometers. So the one degree along equator will be 111 kilometers. Now, obviously, the same one degree on the north or south from the equator will be smaller. Now, if we are talking about close to north pole, this circle is very small, right? And therefore, one third is 360s of this will be significantly smaller. Actually, it goes down to zero. But the biggest is along the equator. Now, as far as minutes are concerned, now minutes is 160s of this. So it's like what? 19 or something kilometers, right? No, not quite. 1.9, right? 1.9 kilometers. So and 160s of this, so that's one minute and one second will be 160s of this, which is well, like 35 meters. So one second along equator is about 35 meters, which is kind of relatively precise. All right. Now, if this is an ideal sphere, then along every meridian we will have more or less the same as we have along the equator. So along the meridian, also one second would be equal something about, like, 35 meters. All right, so that's the position on the surface of the Earth. Now, what about? Well, that actually satisfies the navigation along the sea, right? But how about the air? How about the airplanes? Well, we just add another dimension, which is altitude above the surface of the Earth. So if you want to identify the position of a plane, that's the position on Earth where exactly this plane is above, plus, in addition, an altitude above this surface. So that's the third coordinate, which gives us three-dimensional picture of where exactly the plane is located, or a satellite or something. By the way, you know that some satellites have fixed position above the Earth. So they are always circulating with rotation of the Earth, and they're always above the same point. So basically, you can say that this is fixed position because the projection on Earth is exactly the same and the altitude is about the same, right? So it's a fixed position. In three-dimensional space, within the coordinate system, defined like this. All right. That's basically everything I wanted to talk about. So I would like to pay attention to one little detail. Now, as you realize, we just had two different coordinates, the latitude and longitude, which is defined position on Earth. Now, from the dimensionality standpoint, it means that we have, basically, a surface of a sphere as a two-dimensional space. Same thing as the plane, by the way. So the plane is two-dimensional space because there are only two coordinates which define position. In Cartesian system, it's x and y. In Polar system, it's radius and angle. Now, in this, on a sphere, the surface of a sphere is also two-dimensional because there are only two coordinates which needed to define the position of the sphere. So dimensionality is a very interesting thing. It's actually independent on different ways how you organize this coordinate system. Like the plane, you have two different coordinate systems. We will talk about different dimensions, different systems in three-dimensional space. It will be either Cartesian or spherical or cylindrical, again, three different, and probably there are more. But in any case, any normal good system of coordinates really should be exactly related to dimensionality of the space. So dimensionality of the space doesn't depend on coordinate system. No matter what kind of a reasonable coordinate system we can come up with, it will be always on the surface of a sphere, it will be always two parameters which are needed because the sphere, the surface of a sphere, is two-dimensional space. Okay, that's it. I would suggest you to look at the notes to this particular lecture in Unizor.com and in particular, I also present a few examples of position of certain cities on Earth based on these coordinates, longitude and latitude. And the last thing which I would like to mention is that the real navigation is slightly more complex than this one and it's related to the fact that Earth is not an ideal sphere. And also, there is another very important factor. You see the axis of rotation, north and south pole of rotation, is not exactly the same as magnetic poles because the north on the compass is not exactly directed to geographical north, which means the center of rotation on the axis, but it's slightly shifted and that's supposed to give certain like alternate numbers, numbers which will slightly different than whatever you would consider to be the right numbers using this system. So this is a simplified version. Mathematic is always a simplified version of the nature and so it would be a little bit more complex. There are certain additions, subtractions, which you have to really go through if you really want a system of navigation on Earth. Alright, that's basically. Thank you very much and good luck.