 Hi, I'm Zor. Welcome to Unisor Education. Today we will talk about parameterized curves again as in the previous lecture, but in this case we will talk about, instead of talking about tangential lines, which we have learned in the previous lecture, we will talk about normal so-called line. So, obviously this is part of the course of Advanced Mathematics on Unisor.com. I repeat it every time and I do encourage you to go to this website and which lecture from the website because it has nice notes and it has functionality which you can use like taking example for example. Alright, so we are talking about some kind of a mixture between geometry and calculus. There is a subject called differential geometry where it's actually very, very extensively used. Now, I will just touching the subject. In this particular case we will talk about curve which is given by two functions which are dependent on some parameter. These are functions of the coordinates of every point of this line. So, every point is a function of two coordinates. Each of them are functions of some smooth functions of some parameter t. Where parameter t can be anything. So, t belongs to some kind of an interval, open or close interval, doesn't really matter. And st is going from a to b, then point x, y, where x is a function of t and y is function of t, goes along the curve. That's how we describe the curve. Now it's more universal way of representation than something like y is equal to some kind of a function of x. Because this requires that for every x you have one particular value of y. And curves like for instance circle do not fall into this category because for every x there are two different y's. This one and this one. So, this is not a graph of a function y is equal to f of x. But if I will use some kind of a parameter then something like x is equal to cosine of theta and y is equal to sine of theta, then I will get exactly the same circle of a radius one. But now these are two functions which describe together every point of a circle, the whole thing. So, that's where the parameterized representation is used. Now, we have learned about how to find out the equation of the tangential line. So, if you have some kind of a point x0, y0, x0 is function x. You don't need this. So, x0 is some kind of a function x of the parameter at some t0 and y0 is function y of the same parameter t0. That's how we get into this point. This point is has coordinates x of t and y of t when t is equal to something, right, t0 in this particular case. So, if I would like to have the equation of the tangential line then as we know the general line which goes through the point x0, y0 has this equation where m is a slope and the m is ratio of derivatives at t is equal to t0 of f of y of t and d t is equal to t0 of x of t. So, that's what we have determined in the last lecture. So, take these two derivatives at point t is equal to t0 and their ratio is the slope. I mean, obviously, when I'm talking about ratios like this I presume that the denominator not equal to 0 because if it's equal to 0 it means we have a vertical tangential line which is a completely different story and it's just one particular case, very easy case. I don't want to spend time on this but obviously it's assuming that this is not equal to 0. Now, let's talk about normal. What is normal? Normal is a line which is intersecting the curve at point where it's perpendicular to tangential line. So, you know that if you have a line what is normal? You know that is normal, right? The angle is the same 90 degrees or pi over 2. Now, if you have a curve, now the definition of normal is not so easy, right? So, people have decided that if you would like to be normal to a line which is to a curve what you can do is first you do the you draw a tangential line at this particular point because tangential line resembles to an infinitesimal value the curve at this particular point of tangency, right? So, it's approximately the same as the curve in an immediate neighborhood of this point and now I will just draw a perpendicular to a tangential line and that's what by definition is called a normal to the curve. So, normal to the curve at certain point is basically a perpendicular to a tangential line at that particular point. Okay, once this is defined what do we know about this normal line? Well, number one, it also goes through point x0 y0 so it should have the same type of equation y minus y0 is equal to m times x minus n x0 but instead of m I will put here n not to mix it. So, n would be my slope of a normal. So, m was a slope of a tangent. This is tangent tangential line and now normal would be n. So, my question is what is tangent of this angle which normal makes forms with positive direction of the x-axis and this angle which is a tangential line. Now, this is a perpendicular, right? So, what is this angle relative to this? Well, if this is theta, this angle is 90 degree minus theta, right? And this angle is 100 minus this one, right? Which is theta plus 90 degrees, right? So, we have to really, if we know the tangent of theta, question is how can I find a tangent of theta plus 90 degree and this is the pure trigonometry and now it's easy actually because now we know what we're dealing with. It's a plain trigonometric problem. What we need to know is what is tangent of theta plus 90 degree if we know that tangent of theta is equal to m where m is this ratio, right? Now, that's actually easy. I can actually make a couple of trigonometric transformations to this and find exactly how one is expressed relative to another. Here is how. Well, first of all, I would rather like to have a tangent of 90 minus theta. It's not equal but I would like to transfer it into this. How can I get that? Well, first of all, I have to have minus theta and this is plus theta, right? That's number one. So, I have to reverse the sign. So, this one is equal to minus tangent of minus 90 minus theta, right? Because tangent is an odd function. So, minus in front and minus inside will cancel each other. Now, tangent is a function which has a period of 180 degrees. So, I will add 180 inside. So, I will get minus tangent of, so it's plus 90 degree minus theta. Now, what is 90 minus theta? It's this one. Now, tangent of 90 minus theta, tangent of this angle, which is this gadgetous divided by this gadgetous, is the same as cotangent of this angle, which is this gadgetous divided by this, right? So, it's equal to minus cotangent of theta, right? So, tangent of this is equal to cotangent of this. And the last one, what is the definition of a cotangent? Well, it's 1 over tangent, right? So, it's minus 1 over tangent of theta, which is minus 1 over m, where m is this. So, basically, we have received an equation of the normal. The normal line looks like this. y minus y0 is equal to, so, minus 1 over m. So, it's this one, minus 1 over m, where m is y over x. So, it's minus x over y, right? Times x minus x0. So, that's our equation. Very simple. It's equation of the normal line. So, if you have two functions, x of t and y of t, and you have some kind of a parameter value, t0, which you have fixed, and you would like to know what are the equations of normal and tangential line. Well, these are equations. So, n is equal to minus 1 over m. The tangential line has equation y minus y0 is equal to m times x minus x0, where m is this. And normal line has y minus y0 is equal to minus 1 over m, which is inverted with a minus sign and x minus x0. Well, these are actually the very basic analytical calculations related to curves on the plane. Now, as I was saying, there is the whole subject called differential geometry, which is using these differential kind of calculations to analyze the properties of the curves. But this is a big subject. I just touched the very beginning of this, where actually it starts from. And it starts from the necessity to have the tangential line to a curve and a normal line to a curve. And we basically came up with explicit equation for these two very important lines, which basically characterize the behavior of the function. Another very important behavior of the function is something like curvature. Now, curvature is also expressed in terms of derivatives. And basically, it's something like this, geometrically speaking. This is one curve, and this is, let's say, another curve. It looks like this one is more curvy, if you wish, than this one. And the degree of this curvature can also be measured, and this is just a continuation of this differential geometry subject. It measures tangent, tangential line, normal line, it measures curvature and some other characteristics. All through the implementation of these calculus-style calculations, based on parametric representation of the curve. And obviously, you can parametrically represent not only curves on the plane, but curves in three-dimensional space, which you can still imagine, like, for instance, a spiral, something like this, or helix. Sometimes it's cold. Or in n-dimensional case, which you cannot imagine, which is fine. The only thing which you have with n-dimensional curves is this type of differential calculations. All right, so that's it for today. Thank you very much and good luck.