 Hi, I'm Zor. Welcome to Unisor Education. I would like to finish this story with space coordinates, explaining what is spherical coordinates. Now, this lecture is part of the Advanced Mathematics course for teenagers. It's presented on Unisor.com. The site has lots of notes and for registered students even the whole process of education, including enrolling and exams. So, I suggest you actually to watch this lecture from the site and basically utilize it as the whole educational process. Okay, back to spherical coordinates. Now, we have already covered in the two previous lectures Cartesian coordinates and cylindrical coordinates in three-dimensional space. So, this is the third system which can be used to identify the points in three-dimensional space. So, here is what it is. As always, we have to have some kind of an origin. This is the fixed point relative to which we will try to describe our position of our point using some numerical characteristics. Now, the next thing is the distance from the point in question. So, OA is equal to R. It's a radial distance because this is basically like a radius from the center of origin of coordinates to the point. Okay, now, obviously this alone does not identify the point because anything on a sphere around the origin would have the same characteristic, right? Now, next what we need is we have to have an axis relative to which we will measure the angle of deviation. So, usually it's displayed vertically and you can use the letter Z, actually, again as in Cartesian and cylindrical coordinates to specify this axis as a Z-axis. What is important, actually, from the language perspective, the direction, the vertical direction towards the positive side of this axis is called zenith. So now, you can actually measure the angle of deviation from zenith. It's called polar angle. In some way, it's similar to polar angles on the plane. So, it's deviation from certain directions. This direction is called zenith and this one is deviation from angular deviation from the zenith. Now, does this define our point? Well, obviously, again, not exactly because any point which has the same angle with my axis, with my Z-axis would qualify. So, it's like a cone, basically, right? So, this point A, if R is fixed, then you have a sphere, right? And if phi angle, polar angle, is fixed, then you have a cone and intersection between a cone and a sphere would be a circle. So, it's still not exactly the determination of the position of the point. But, well, we need three coordinates. You remember, right? In three-dimensional space, we have defined only two of them. So, this is angle ZOA. This is phi. Now, so, how can we calculate the third coordinate in spherical system of coordinates? So, we need a reference plane, which is somehow equivalent to X, Y plane in a Cartesian coordinate because it's perpendicular to the Z-axis and it goes, obviously, through origin. So, the same plane we had in the cylindrical coordinates, Cartesian coordinates, and here we have it in spherical coordinates. Now, what do we do with this plane? Well, obviously, we will project our point and we will have this direction from our origin to a projection and measure the angle of deviation, again, angle of deviation from some axis which we, again, fixed on this plane. So, the angle in this case, for instance, in this case, it would be this one. Now, here is a slight difference, I would say. Angle phi is an angle of deviation from vertical, from positive direction of the Z-axis towards our point and it can be, obviously, from 0 to no longer, not greater than 180 degrees, not greater than pi, right? And it's always positive. Now, this is also positive, but the angle can be from 0 to 360 degrees, right? Because we're always measuring counterclockwise if you look from the positive direction of the Z and it will be always from 0 to 360. So, now, we have, and we can actually put letter x because in some way it's equivalent to the x-axis in the Cartesian coordinates. In many cases, picture even contains the third axis, the y-axis, which is not really used in spherical coordinates because we are using only the x-axis to measure the deviation from the positive direction of x. And, by the way, it's called azimuth. So, now, we are talking about angle x of a projection and that's usually the letter theta, Greek letter. Now, as far as letters are concerned, I think this is more traditional in mathematics. In physics, they prefer, actually, to use the theta for a polar angle and phi for azimuth on the reference plane. Well, I mean, letter is letter whatever you want to use, you're obviously free to do whatever you want. All right, so this is an explanation of what is spherical system of coordinates. So, it's three characteristics. Distance from the origin, oa, angle which ray oa is deviated from positive direction of the z-axis from the zines. So, that's phi. And then, if you drop the projection from a onto the plane of reference, which is perpendicular to z-axis and goes through origin, obviously, on which there is a chosen direction, a positive direction. So, the deviation from this chosen fixed direction is an azimuth that's the third characteristic. So, again, what do we need? We need origin, we need z-axis, we need reference plane, and we need an x-axis from which we count the azimuth, right? So, these are three characteristics, well, three construction elements which we need to be able to determine spherical coordinates. Now, let's spend some time and talk about how the point defines spherical characteristics and how spherical characteristics define the point. Again, it's supposed to be one-to-one correspondence, right? And, by the way, what I also mentioned is that phi is from zero when we are actually at zines no end to phi. Phi would be when it's vertically down. So, the A can be here or A can be here. So, this is zero phi and this is pi, 180 degrees pi. Now, as far as theta is concerned, that would be obviously from zero to no... to less than 2 pi. Now, this is less because I don't want to specify 2 pi because it will correspond to the same as zero, so we don't really need it. Well, obviously, r is supposed to be from zero less than infinity, only positive. So, these three determine the point and the point determines these three. So, let's just follow this logic. If I have a point and I have these axes and reference plane and the x-axis, etc., how can I calculate the three spherical coordinates? Well, obviously, well, I forgot to mention, unit of measurement, unit through length, right? So, you measure OA, that's your r, then, since you have OA and you have this axis, z-axis, using these two lines, we draw a plane and within that plane, our phi angle is just a flat angle which we can measure at angular units, regions or degrees, right? So, that's how we measure phi. Just have a plane through OA and z-axis. They are intersecting, so there is one and only one plane, so no problems there. Now, and the third characteristic, you have to project A to the reference, plane and take the OAP ray and compare direction with the positive direction of the x-axis and that would give you the azimuth. Measure the angle counterclockwise from zero to no more than two p. Okay, that's how we go from a point to three coordinates. Let's go back. So, let's say we have these three coordinates. What do we do with them to construct a point? Well, using r, you can construct a sphere. That would be a locus of all points which are on the same distance r from the O. Among these points, there is our point A. So, it defines a surface on which our point is located. Now, since I have angle phi, I can create a conical surface with the generator being on an angle phi from the axis of symmetry of this cone, right? That would be like a cone like this. This is a cone on one hand. On the other hand, we have a sphere, right? A sphere would be something like this. So, the r will give me the sphere. The phi will give me a cone and that's the curve where the cone intersects the sphere. So, we have a two-dimensional sphere after we use one coordinate. Then we have line, curve, whatever which is kind of one-dimensional which specifies even further. So, our point A is not just on the sphere but on this particular circle on the surface of a sphere. And the third coordinate since we have the azimuth, the way how we do it, let's project this circle onto our reference plane. I will get a circle here on the plane itself, right? And since I have it on the plane, I can basically have this angle and construct a ray and wherever this ray intersects, that would be my projection. Now we go all the way up back to the sphere or a circle and we will get the point A. Just draw perpendicular to the reference plane from this point where my ray intersects this circle. Okay, so this is how we go from numbers to points. And the last thing is I would like to present a couple of examples. Spherical coordinates seem a little bit more complicated than cylindrical. Cylindrical may be a little more complicated than cartesian, but there is a purpose in everything. Now what's the purpose of spherical coordinates? Well, actually my third example would probably explain it a little better, but let me start from two trivial examples. Now the first example is a sphere. I would like to present an equation which represents a set of points which lie on the same sphere of a radius r. So what would be that equation? That's the equation. If r is given and this is the variable coordinate, one of these coordinates, then this is I mean it cannot be simpler than that. So sphere, however complex that thing actually is, this geometrical object is, its description using algebraic equation in spherical coordinate is simple, right? Now, how would it look in cartesian coordinates? Well, that's this way. I think we did a couple of times examine this, but this is a distance. If x, y and z are coordinates, then using twice the Pythagorean theorem you will get this as a distance from the origin of coordinate. So using this Pythagorean theorem twice, that's the equation you will get. This is simpler than that, obviously, right? Now, another example is, let's say you have a cone, but not just cone. I would rather say conical surface, okay? Conical surface with epics being at the center of the origin of coordinate. So this is my conical surface, okay? Infinite in both directions. So there is somewhere a director's, right? There is epics coincides with origin of coordinates. Now the plane where my director's is is supposed to be perpendicular to the z-axis, right? And the center is supposed to be on the z-axis, right? Center of this director. So director's is given, epics is given, right? So all the lines which go through epics and one of the points here they are for the cylindrical surface. Now, what's the equation of all these lines in the cylindrical surface? Well, if this is angle phi between the axis of this conical surface and the generator's, then that would be a direction. I mean, sorry, if this is alpha. So if I would like to express a conical surface with around the z-axis with the center of the origin with an angle alpha between the axis of rotation and the generator's, then this is the equation. So phi is a coordinate. We don't fix r, we don't fix theta. So everything is just defined exactly by one simple equation. Now the third example is a little bit more complicated and a little bit not mathematical. It's related to position of a star. So you know that astronomers are usually telling you if you want to see some specific star at this particular location on the surface of the Earth, at this particular time you have to direct your telescope at this particular angle to the horizon, something like this, right? Now, what does it actually mean? Well, it actually means a convoluted way of presenting spherical coordinates because here is what I can suggest. So let's say this is a star. Now this is Earth. Now the Earth actually has an axis of rotation, right? And you can always display it as vertical axis, right? It doesn't really matter. So you are somewhere on the surface of the Earth. Now obviously it doesn't really matter where exactly you are whenever the surface is rotating you will be basically on a parallel, right? perpendicular to the axis of rotation. Now, from the position of a star actually this is still a very small object and the angle between axis of rotation of the Earth and direction on the star this angle is actually the same. However, whenever you are rotating you see the star actually as it rotates on the sky, right? So how to measure where exactly this... So if you consider that the Earth is standing still, right? And relatively to Earth your star is rotating, right? on the sky. So you see it at different places in the sky. What you do is you can project the position of the star onto the reference plane where you are. So it goes through the parallel where you are perpendicularly to the axis of rotation. Then you can obviously project this rotation of the star relative to some direction. Let's say you have direction to the north within this plane. You have direction to the north so you can always measure this angle. So whenever astronomers are saying that you can observe your star at certain angle from the horizon it's this angle, right? Which is 90° minus this angle. So it's exactly the same thing just a different angle. So instead of... instead of angle phi you are using angle 90 minus phi, right? 90° minus phi. Alright, so that's the same. Now the second thing is they're talking about specific time. Now what time means? The rotation of the Earth actually. So the time is the number of hours from certain moment in time when the Earth actually... when zero actually is... when the Earth goes through the Greenwich Meridian, right? So whatever the time specified is is actually a measure of the azimuth which you should observe your star. So it's just a little bit more complicated to talk about azimuth and polar angle. It's simpler for the consumers to talk about the time at a specific location on Earth and the angle from the horizon. But in theory it's exactly the same type of things. Now we don't actually talk about individual distance because that's not really what's important. I mean, yes, obviously this is the third coordinate. But our purpose, our astronomical purpose is just to have a direction. So the phi and theta are basically defining this direction and instead of phi they're using 90° minus phi and instead of theta of time. Which is actually a measurement of the azimuth considering the speed of rotation of the Earth. Okay, so that's it. That's all I wanted to talk about spherical coordinates. I doubt you will be using this in your practical life unless you might be an astronomer or something like that. But still it's interesting and what also is interesting is that no matter which coordinate system you're using we have described three of them Cartesian, cylindrical and spherical. And I'm sure anybody can think about some other system of coordinates. But what's interesting is that in the three-dimensional space where we live it's exactly three numerical characteristics are needed. No matter how you define your system of coordinates. In the Cartesian system you need three linear dimensions along the three axes X, Y and Z. In a cylindrical system you need two linear and one angular dimensions, right? So you need linear distance to a projection you need an angle vertical also linear. And in spherical system you have one linear and two angular. So no matter how you design your system you still need three numerical characteristics. And that's what actually making our space three-dimensional. This is a property of the space and no matter how you try to position your point, to define the position of your point in the three-dimensional space you need three coordinates, three numerical characteristics. So being of a certain dimension or dimensionality of the space is really much deeper property of the space than anything we just associate with any coordinate system. Alright, that's it. I would recommend you to go through the notes for this lecture. Maybe I missed something I'm not sure. And I do recommend you to actually go to this website and engage more actively in the whole process, in the whole educational process from the beginning, from the first lectures about numerical systems, etc. down all the way. That's it. Thank you very much and good luck.