 Hi, I'm Zor. Welcome to Unizor Education. Continuing talking about graphs, I would like to devote this particular lecture to a completely different way of graphing dependency between two variables on the plane. We used to have rectangular or cartesian as they're called sometimes, covenants on the plane. So you have one axis, which is x-axis, another perpendicular to it y-axis. Both are linear dimensions basically. They are represented the points where any number is basically represented by a certain point on this axis and point on this axis and something like the dependency which you have is a set of all points, x, y, on this particular plane where x is projection to the x-axis and y is projection to the y-axis and this particular equation is satisfied. So this is the rectangular or cartesian system of covenants. Now, I would like to introduce a completely different approach to represent a point. Here it is. Let's just assume that the position on the plane is not defined by a relative position to two perpendicular axes, but you have only one axis and it's not really an axis, it's an array. I mean it can be called an axis, but it's only half line. Now, every position on the plane can be characterized by two things. Let's draw this particular vector to this point. So this is called pole, this point pole. This is called polar axis. This I'm not sure that it can be called as a polar vector if you wish and this angle can be called polar angle. Now, position of this point is defined by how much this particular vector in angular dimensions is rotated from this position along the polar axis. So if you take this particular direction, how much do you have to turn to get to this particular direction? Now, obviously for any point on the plane, there is this particular angle which is basically from 0 to 360 degrees or from 0 to 2 pi in gradients and the angle defines direction towards the point. Now, how far this point is from the center is a second variable which we need which is basically the distance from the center. So two things. We have a distance from the center which is usually denoted by letter r and the angle from the polar axis angle of the vector two words our point from the polar axis. It's usually denoted by letters either phi or theta. I'll use probably phi, it doesn't really matter. In my notes I think I'm using theta. Both are equally acceptable. Obviously you can use anything but this is this tradition. The r and phi or r and theta are traditionally for these two things. So knowing these two variables is sufficient to determine the position of the point on the plane. So instead of two perpendicular axis and y axis we are using the polar axis there are still two different variables which we need to determine the position of any point relative to the polar axis. So let's just give a couple of examples. What would be coordinate of this point on the polar axis itself on the distance let's say a. Well coordinates would be r is equal to a and phi is equal to 0. Since we are on this polar axis we don't have to move that's why phi is equal to 0 and the distance we have to move from the pole itself is equal to the a as I was just saying so r is equal to a and phi is equal to 0. So coordinates at this point are a 0. Okay how about this point I shouldn't say minus. I'm talking about distance so this is also distance a so the continuation of the polar axis to the left from the pole to the opposite direction I shouldn't again say left because there is no left and right. This is a positive direction and that's the only we have. So I'm going to the negative direction against the direction of the polar axis also on the distance a. So what would be coordinates of this particular point well again the distance is still a how about the angle but from this position we have to turn to this. So it's 180 degree or usually it's in regions which is phi. By the way I'm using gradients, angles and other vectors and other terminology here well obviously you understand that we are talking about graphs we are talking about something which is in the middle between algebra and geometry so that's why I'm using for instance the region measure of the angle if you're not familiar with this you have to go to geometry lecture which where I explained what actually region measure of the angle is so I will use it basically as it is a known term and hopefully you do know if you don't there is a reference the same Unisor.com contains the lecture about the angles and how they are supposed to be measured. So as I was saying coordinate of this point are same a note this is a distance which is always positive there is no negative direction on this particular axis this is only one positive direction so everything else is achieved through turning in this case by pi so that's coordinate of this. How about perpendicular angle on the distance a well let's think about the distance again is a and what's the angle? angle is 90 degree or half of the particle so this is a pi over 2 and on this side if it's also distance a same thing r is equal to a well we can actually turn 3 quarters of the full circle but this is not again traditional measurements. Traditional measurements is going to the opposite direction by a quarter of a circle so anything which is on this half of the plane is usually measured in a negative value of the angle because the positive value is counter clockwise as we know again that's the geometry measure of the angle is always positive when you're moving to the counter clockwise direction. Moving from here to here is clockwise and that's why it's negative so the coordinate of this particular angle is again r is equal to a and pi is equal to minus p over 2 so that's the coordinates of this one at this point and you know if it's something like on a bisector of this angle at 45 degrees or p over 4 it will be correspondingly a and p over 4 etc. So this is how the polar system of coordinates is built again we have the angle and we have the distance from the origin from the pole. Well let's do a couple of examples well example number one is r is equal to is this a function remember the function I'm not using x and y anymore the function would be described as this and again I shouldn't really say a function a dependency between r and pi is expressed in this particular way is this an example of this dependency yes absolutely it means pi can be anything so no matter what it is but r minus a is equal to 0 because that's what it is this is our function so for any phi for any angle any point which is distanced from the pole pi a would satisfy our our equation now obviously a should be greater or equal to 0 because r is a distance right so only equation of this type or this type where a is non-negative makes sense so let's draw the graph of this particular equation well this is our pole this is our polar axis and what we are saying is that any point on a distance a which is this point this point regardless of the angle regardless of the angle any point which is distanced by a from the pole would lie on the graph of this particular equation so what is it well obviously this is a certain so this is an equation of the circle in polar coordinates now recall that the equation of the circle in Cartesian coordinates is x squared plus y squared is equal to x a little bit more complex right than this one so circle is much more simply represented in the polar coordinates than in Cartesian well another example by the way this is a circle with a center coinciding with the pole itself of the polar coordinate system if center is not coinciding the equation is much more complex but let's not go into this we are talking about just very simple things now how about straight line which is going through the pole well let's just consider any point on this line has exactly the same angle phi in this particular case any point on this piece of the line this is the same angle plus 5 plus 180 degree so it's a combination of two pieces one piece is determined by phi is equal to let's say alpha this is an angle alpha and I presume that alpha is from 0 to to phi and this piece is phi is equal to alpha plus phi so union of these two graphs basically of these two graphs represents the whole line I mean this one represents half of the line and this one represents another half of the line so how to unionize well the previous lecture I was talking about unionize two different graphs you can use for instance the product equal to 0 right so phi minus alpha times phi minus alpha minus phi equals to 0 this might be an equation of the line r is not part of this r doesn't depend on r it depends on the phi so the phi can be either equal to alpha or equal to alpha plus phi so for any alpha within this range that's true last example which I wanted to present is this a spiral how can I draw a spiral look I'm talking about right now the spiral which is called our comedians spiral for Greek mathematician archived I don't know how to pronounce well in any case as you see I'm starting in the beginning so when my phi is equal to 0 my r is equal to 0 then as phi increasing my r is also increasing so as I move my vector this way I'm moving away from the center from the pole and that's what actually makes the spiral whenever I make a one circle in this particular case phi is equal to basically 360 degrees right 2 pi so if I have an equation which looks like this what does it mean well again if phi is equal to 0 r is equal to 0 then my phi increasing my angle increasing so I'm going this way and my r is increasing by the same thing now by the end of one circle my phi is equal to 2 pi right 1 circle 360 degrees so this particular segment is equal to 2 pi in lengths but then I can increase my even further now when it goes to second rotation I will have point here which is 4 pi now 2 pi and again 2 pi so it's growing this way this is how the spiral actually is increasing in ranges so this is called the Archimedes spiral we can actually have it a little bit more complicated instead of just playing r is equal to phi let's see what happens if I have a slightly more interesting equation if I have a linear dependency r is equal to a plus b phi what's this well let's think about let's just draw this we start again from our polar axis so at phi is equal to 0 my r is equal to 8 so h should be greater than 0 so this is 8 now let's consider that b is positive so as phi is increasing r is also increasing so it will be further so if I will draw let me just get a little bit smaller scale here what if a is here this is a ok so if I will start from this this is the point with phi is equal to 0 r is equal to 8 now my increasing phi if b is greater than 0 I'm moving further from 8 so if this is my circle of the radius a my point would be further from this circle and further and further and further again and again so I will increase the radius for positive b as phi is increasing if b is negative what's interesting is then as phi is increasing I'm subtracting basically something from 8 so instead of going outside of this circle of the radius 8 I'm going inside so it goes this way closer and closer and closer to 0 and at some point I will reach r is equal to 0 when phi is equal to if this is 0 so it's minus a over b right so if b is negative a is positive so that's why it's a positive way it's minus and b is negative so it's positive but this is the maximum value of phi which I can afford because at this particular value r is equal to 0 and my graph goes to the center of the pole and there is nowhere else to go r cannot be negative r cannot be negative so for a negative b my spiral is going inside and basically going tighter and tighter around 0 around point 0 and finally it does reach in a finite actually number of rotations if b is positive it goes outside of the circle bigger and bigger and bigger and that's to infinity alright so that's basically what the spiral is the Archimedean spiral is it's just an interesting example of how simple certain very complicated curve and this spiral is a complicated curve how simple it can be expressed in polar coordinates something which is simple in Cartesian coordinates like for instance the equation for the line it's just a linear function between x and y is a little bit more complex in the polar coordinate I have to basically combine two pieces together etc and vice versa something which is simple in polar coordinates like for instance Archimedean spiral is a very difficult curve in the Cartesian coordinates so that's why people are using both at certain you know different cases by the way speaking about polar coordinates and it's usage actually there is an equivalent in a three-dimensional space we are using as you know the longitude and latitude on the surface of the herbs basically these are spherical coordinates which is kind of extension of the polar coordinates to three-dimensional space alright so what's next is we would like to see how polar coordinates are related to Cartesian coordinates in some simple case well let's consider you have Cartesian coordinates and polar coordinate which is with a polar axis coinciding with x axis because that's the simplest way right so the polar coordinates are on the same plane this beginning of the Cartesian coordinates is the pole and the positive direction of the x axis is the polar axis and now I have a point which has on one hand coordinates r and phi in a polar coordinate system and x and y in the Cartesian coordinate system the same point on the plane now my task is to express one set of coordinates through the other well now again we are talking about graphs so it's in geometry and algebra in this case it's trigonometry which is kind of extension of geometry towards algebra so I will use trigonometry and again I will use it as is I would basically urge you to go to the corresponding lectures or whatever the educational material you have to learn about it but I will use trigonometry as given so let's just draw the perpendicular here so this is x and this is y right because the coordinates are x y in Cartesian system now this is r and this is phi so let's just consider this particular picture how to express let's say x y in terms of r and phi well basically because x is equal to r cosine of phi and y is equal to r sine of phi basically it's from the definition of the sine and cosine almost now how to do it in reverse how to find out what my r and phi are if we know x and y well r is easy because this is the Pythagorean theorem right so since r square is equal to x square plus y square r is equal to square root of this arithmetic square root positive value x square and y square are also positive so it's always correctly defined regardless whether the point is here or here or here or here with a phi is a little bit more difficult here is y basically from the definition of tangents which is another function of trigonometry tangent of phi is equal to y over x this is practically from the definition most at least for the phi within the range from 0 to 90 degree to p p over 2 that's basically true so from this we can always find phi using an inverse function called arc time so this is I would say the beginning of the process of pressing r and phi well actually the phi I already finished the phi because now we have some problems first of all in this particular case x cannot be equal to 0 right so what if x is equal to 0 but let's just think about this x is equal to 0 is a special case it means all the points where x is equal to 0 are here right so for all these points phi is either equal to pi over 2 90 degree or minus pi over 2 minus 90 degree right if point is here it's 90 degree pi over 2 if point is in this particular part of the y axis the angle phi is equal to minus p over 2 so I can actually consider all these cases so if x is equal to 0 if y is equal to greater or equal to 0 then phi is equal to pi over 2 if y is less than 0 then phi is equal to minus pi over 2 so we have finished case x is equal to 0 now we can define we can divide y over x whenever we want to and we will have no problems now what if x is now greater than 0 or less than 0 in this particular part or in that particular part then it's easy it's only one case and we can say this formula actually is correct arc tan y over x now why is it correct if y is also positive then it would be positive value and we will have the positive value of phi from 0 to pi over 2 if y is negative the result would be symmetrical but with a minus sign and that actually corresponds exactly to this picture so if x is less than 0 that actually presents a little problem if less than 0 it means it's here so the points are either here or here so let's think about if point is here then this angle is the same as this angle plus pi so I can always say that if y is positive for 0 then my phi is equal to arc tan y over x plus pi and finally if x is less than 0 and y is also less than 0 so it's somewhere here I can always say that this is the same as this plus pi I mean sorry we have to go negative, negative pi from this point we go backwards because all these guys have negative phi so it's equal to the same arc tan y over x minus pi so it might actually look a little complicated but this is the main thing so as soon as you have defined it for the first quadrant of the coordinate plane everything else is just basically mechanics but the problem is that you have to know a little bit trigonometry to convert one to another and basically that's it I mean if you know just a little bit of trigonometry then the conversion will be very easy so this is conversion from polar to cartesian and this is a conversion from cartesian to polar, r is very simple but phi is, it depends considering many different cases so basically this is all I wanted to talk about the polar system of coordinates again as I was saying under certain circumstances polar coordinates are more convenient than the cartesian however I should really say that cartesian are much more often used in start of mathematics but it's interesting because some very nice curves like archimedean curve can be expressed, spiral can be expressed in this particular way alright thank you very much and good luck