 Hi, I'm Zor. Welcome to a new Zor education. We continue talking about solid geometry. I'm introducing certain concepts which will be used in all the future materials of theoretical kind and problems. So this is the lecture which is about conical surfaces. I would recommend to view this lecture from Unizor.com website because it has side notes which basically can serve as a textbook. Also, I'm planning to put a lot of materials into examination part of this website, Unizor.com. There is a lot of exams already in this course, especially about algebra and geometry on the plane. And I am planning to put basically the exams for every subject which I'm touching in this course, course of advanced math for teenagers. The only thing is for exams you have to just register. The site is free, so registration is free anyway. Alright, so I'm trying to introduce certain concepts in solid geometry and today's topic is conical surfaces. I assume you are already familiar with the previous lectures which are introducing certain other elements of solid geometry, which are points, lines, planes, cylindrical surfaces. So right now it's basically a continuation, different kind of surfaces. If you are familiar with the lecture which explains what cylindrical surfaces is, this would be very much similar to this. I'm just introducing a concept of a conical surface. So what is a conical surface? Let's assume you have a space curve, whatever space curve is. Now, I'm drawing this on the board, which means this particular curve is flat. In theory, in solid geometry, this is a three-dimensional geometry, I do not have this requirement, so it's any kind of a curve in the space. So it can be whatever, helix for instance. It doesn't really matter. So let's assume we have this curve in space. Let's call it C. We also assume that somewhere else we have a point, fixed point, let's call it S. Now, what I'm going to do now is the following. Every point on this curve I will connect with a straight line with this fixed point S. So all these points on all these straight lines, they are actually forming some kind of a surface. And, well, that's it. This surface is called a conical surface. Now, it's infinite in both directions from this point S, because these are straight lines. They are infinite in both directions. Now, this line, this curve, which is used to connect the points which are connecting to the fixed point S, this called directrix, very much similar to cylindrical surface, if you remember it. Now, this fixed point S is the new one. It's called apex. Well, sometimes it's called vertex, but vertex is a more general name. I would prefer to call this apex or apex. So, these are just two different words, terminology about the conical surface. And basically, that's it, which defines very, very general conical surfaces. Now, let's just consider different cases which can occur. Let's assume first that we have this curve C, the directrix. Let's assume it belongs to some kind of a plane. So, I'm introducing a side view on this plane, and let's just assume that this is a curve. So, we assume that not only this is a flat curve, which means it belongs to one particular plane. Let's also assume that this is just a closed curve. So, it basically encloses some kind of area on this plane. And let's also assume that the point S is somewhere outside of the plane. Now, if I will connect every point, it's infinitely... Okay. Now, in this particular case, it's appropriate to call this area inside the curve on the plane a base of the conical surface. So, that's just another kind of terminology which I would like to use sometimes. Now, as you understand, the conical surface has basically two independent halves, if you wish. This piece which connects to the curve and further, and those pieces of the straight lines which are going beyond the point S. Well, that's what it is. In practical situations, we usually will be considering this case when the director is a flat curve. Usually, it's some kind of a closed curve. And in most cases, we will not consider an entire conical surface, but only from this point to this plane, this piece, without these infinite continuations of it. Now, why we will be considering these? Because this is now some kind of a geometrical object which can have, you know, volume, area, whatever are the characteristics, the altitude, etc. That's what we will be dealing with. However, again, you should understand that it's just a piece of a conical surface which is actually infinite in both directions. Now, let's just consider a couple of interesting cases. Now, the one case which is, I would say, trivial, and most likely we will never be dealing with this particular case, but I would like actually to present it as an example of a conical surface. So, let's assume that our director is a straight line. So, it's in space, but it's a straight line. And let's assume that there is something outside the point S, the apex, which is outside of this line. Now, what would be the conical surface in this case? Well, let's just try to connect each point on this line with apex. This is one point. Now, what actually will be the set of all the points on all these lines which are connecting apex S with each point on the director's? Well, obviously it will be the plane which passes through this line and this point. So, in this case, a conical surface is a plane, however strangely it sounds. Now, another example, also kind of strange, I would say, but nevertheless. Let's assume that our director is a flat circle, a circle which is lying on some plane. Let's consider that this board is the plane. So, this is my circle. So, this is director's. And let's assume that apex is right in the middle of it in the same plane. Now, what happens in this particular case with a conical surface? Well, again, let's just connect each point on the circle with its center and what will be? These are the lines and what will be a set of all these lines? Well, it will be this particular plane where the circle belongs to. So, again, the plane would be a conical surface. Now, these two cases are not really typical and we will never return to these cases again. I just wanted to present them as a little bit unusual conical surfaces. Now, what is a usual conical surfaces? Okay, let's consider it this way. Let's assume that we have a plane and we have a circle in the plane. Now, the circle looking from the side looks like an ellipse anyway. Now, let's also assume that you have a point S above this plane such that its projection or perpendicular to this plane projects to the center of the circle. Now, what we will do? Now, we will connect each point, something like this. So, each point on the circle is connected with a straight line to the apex S. Now, this is a very familiar to everybody conical surface. It's not yet, well, sometimes it is called a cone. Sometimes only half of this from this point down is called a cone. And sometimes only the piece from this apex S to this plane is called a cone. So, whatever the context is, so the cone is something which is one of these three different things. It's either both infinitely directed pieces of this conical surface, only half of it sometimes is called a cone and only a piece from the apex to the plane is called a cone. Most likely, we will be dealing only with this piece from S to this plane and we will call it a cone. So, that's another example of the conical surface which we will be addressing in a separate lecture, the cones. Now, and one more thing is again related to a director's which is on the plane. And let's assume we have a polygon, something like this. And the apex S projects from above this plane to some point inside the polygon, usually it's inside. Yeah, we can also consider when it's outside but usually it's inside. And that would be in this particular case, if I will connect all points to this particular thing. Well, now this is another very familiar piece. If you consider this conical surface only from this to this, it will be something which is called pyramid, which we will also discuss in a separate lecture. So, my purpose of this lecture was to introduce a conical surface as a foundation for certain other objects in solid geometry like pyramids and cones, which we will be really talking about. Most likely, we will not return to the concept of a conical surface at all. We will be dealing with particular cases of conical surface. But again, my usual approach is to introduce something general and then from that general consider specific cases. So, basically that's it for today. Oh, yes, I wanted to mention one more thing. If you have a conical surface, you actually can flatten it, which means, for instance, if it's a pyramid, you can cut it along the edge and open up and it will be flat on some kind of a plane without stretching. Now, if you have a surface which is not like conical, if you remember the cylindrical surface also has this property. If you can flatten it without stretching or any kind of distortions. If it's necessary, you can cut it along the edge, but anyway after that you can just open it up and flatten it completely. Why? Because it consists of lines actually. This also consists of lines and that's why you can actually cut it, open it and it will be flat without distortions. So, that's one thing. And what else? And the last thing which I wanted to mention is, it basically depends on how you view the conical surface. You can view it as the surface which is formed by all the points of all the lines which are connecting S to every line on the director's. Or, alternatively, you can view it as movement, result of the movement of one particular line which connects S to one particular point. And then you move this point and with the moving of the point, the line is moving and whatever this line actually, as a result of its moving, whatever points it's going through, that would be a conical surface. It's actually absolutely equivalent kind of a view. That would be probably the last thing which I wanted to mention about the conical surfaces. I do suggest you to maybe read the notes for this lecture on Unizor.com. Well, because when you are reading something which you have already heard about, it better actually inculcates in your minds. So, that's it for today. Thank you very much and good luck.