 Hi, I'm Zor. Welcome to Unisor Education. I would like to continue introducing certain concepts of solid geometry. This is part of the course of advanced mathematics for teenagers presented on Unisor.com where I actually suggest you to watch this lecture because the lecture itself on this website is accompanied by notes, very important. So I'm just introducing certain concepts without proving any theorems, without solving any problems. So this is just subjects of solid geometry or objects of solid geometry. What exactly we are dealing with, what kind of geometrical figures we are dealing with. Now I have already introduced points, lines and planes together with certain axioms. Today I am going to talk about cylindrical surfaces. Again, just introduction, what basically this is all about without any kind of properties, proving, etc. So now the term cylindrical surface obviously implies the word cylinder, which obviously is familiar to everybody, right? So everybody knows this is a cylinder, right? Now cylinder is actually one particular object which is definitely will be learned and properties will be discussed in this course. But I would like to say that this is just one of the different set of different surfaces, which all together are called cylindrical surfaces. Why am I starting with cylindrical surfaces rather than just define what cylinder is all about? Well, it's all because, you know, mathematics is all about abstractions, generalizations, etc. So it's always it has always been my principle to introduce some general concept and then derive from that general concept a particular concrete object to study. We will definitely study the cylinders and probably no other cylindrical surface will be studied in any details. But I would like just to introduce the concept so you will familiarize yourself that there are basically much more general categories which we derive concepts from. Okay, so the general category is cylindrical surface and I'm going to introduce it right now. Let's imagine that somewhere in space there is a curve. Now I will draw this curve on this particular board, which is plain actually, but it should not really imply that the curve is really flat on this particular plane. The curve can be whatever you would like in space. Let's say this point is tracing the curve. The curve can be something like this, which is completely outside of any particular plane. It's completely in three-dimensional space. So there is this curve. Let's call it C. Now at the same time somewhere in space there is a straight line. We call it B. Again, this straight line lies in the same plane as this particular curve, but that's only on my picture because I cannot make a picture in space. We need holographic images, etc. I don't have this technology. So there is some straight line in space. Now let's consider the following. Through each point on this curve, we draw a line which is parallel to this line like this through each point. Now all the points which belong to all these lines parallel to this one form a surface, some kind of a surface. Now basically this is by definition a cylindrical surface. So a cylindrical surface is a set of points which belong to all the different parallel lines, the parallel lines to this one and passing through all the points on our curve, wherever this curve is and whatever this curve is. So that's basically definition. Now what kind of examples we can really imagine? Well, let's consider it this way. I will just give you two examples. The example number one, let's consider that line C is a straight line, but this straight line is in space and it's not on the same plane as line D. So basically something like this. If this is the D, this can be line C. Non-parallel, not lying in the same plane, completely different. Now imagine that through each of these points on this straight line C, I have a line which is parallel to this one. What will be the resulting surface? Well, the resulting surface will be obviously some kind of a plane and since each line is parallel to this one, it will be actually the plane parallel to this line. By the way, I have not defined in strict terms what is a line which is parallel to a plane. I basically resort to some intuitive understanding of this. Well, basically if the line is parallel to a plane, it never intersects the plane. So everybody understands intuitively and we will delay the rigorous definition to further lectures. But anyway, this is an example. So the plane can be a cylindrical surface according to this definition if this line is a straight line which is not lying in the same plane with this one. And let's consider another example. Another example is let's say we have a circle lying in the plane which is perpendicular to this line. So that's why I draw an oval, not a circle because we are looking from the side. So you consider that this is basically the plane where our circle is drawn. And this plane is perpendicular to this line and again I resort to your intuitive understanding of the term perpendicular. Now what happens in this case? Well, in this case we will have parallel lines which are going through this. Right? So what will be the result? The surface will be a cylindrical surface and actually this is the closest we can get to something which we all understand what cylinder is about. The only thing is this is an infinite cylinder. It continues up and down to infinity because the whole straight line actually participating. Now from just viewing point you can really imagine this not as a set of this surface, not as a set of points which belong to all the lines parallel to this one and drawn through each point on the curve. You can actually consider it slightly different. Let's take one particular point, draw one particular line parallel to this one and then we will move this line along the curve basically. So it's exactly the same thing. So you can imagine this cylindrical surface as a trace of a straight line which is drawn through a point on the curve and then the point is moving. And that's how our line is moving and all the points which this line actually is crossing belong to this particular surface. Now from the terminology standpoint let's introduce some terminology. Now this curve C is called directories. Why? Because it actually directs the position of every straight line which goes through it. So or if we're approaching this from the moving straight line purpose so it directs where it moves actually. It moves along this curve. Now this line which every one of these lines supposed to be parallel to line D is called generatrix. Well obviously from the word generate because this line helps to generate the surface because every line I'm talking about is supposed to be parallel to this. So directories or directories and generatrix. Now each line which constitutes a piece of the surface sometimes is called ruling but I'm not sure ruling. I'm not sure this is a kind of a general terminology. Sometimes each line which basically is part of the surface can be called a ruling. One particular ruling. So that's as far as terminology is concerned. Now what's interesting about cylindrical surface? Well in most of the cases maybe even just in all the cases I have to think about whether it's all the cases but it's certainly true for all the plane cases. You see if this is a cylindrical surface something like this it's always possible to flatten it. Just open it up basically. Probably regardless of this curve although I'm not a hundred percent sure maybe there are some very very complicated curves and if we will move along this curve maybe we will have some very tricky surface. Some intersecting with itself etc. There are some details but in most plane cases when the curve along which we are moving this directories if it's relatively simple then our surface will be relatively simple. So this is a cylindrical surface. This is a cylindrical surface. This is a cylindrical surface. So all these kinds of simple surfaces simple cylindrical surfaces can be flattened without stretching without cutting etc. Now by the way which is not the case with certain other surfaces in space. For instance if you will take a spherical surface like the surface of our earth for instance almost spherical you cannot flatten it without any kind of stretching and distortions. That's why the map which is presented on the surface on the flat surface of the page of the book for instance is not really an exact representation of the real surface of the earth. It's really distorted in some way. So they're usually choosing some small part which is presented really accurately and then something like north and south poles are really distorted. That's because the sphere is not a cylindrical it's a spherical surface and we will talk about this in some other lecture. So flattening of the of the cylindrical surface is just one of the um properties which we can talk about. So basically that's it I just wanted to introduce the concept of the cylindrical surface and again in the future we will use this concept for probably only for one particular case. We will use the case um with um so-called right spherical cylinder when our directories is um a circle on some plane and um our directories is um our the plane where our directories is located is is perpendicular. Let's put it somewhere like this. It's perpendicular to the generator. So in this case whatever we will get forming a cylinder a cylinder cylindrical surface would be something which is called the right spherical cylinder right because it's perpendicular. This plane is perpendicular to the generator and circular because the directories is is a circle. So we will have the real cylinder something like this. This is invisible part of it and obviously a true cylindrical surface would be infinite in both cases. The cylinders which we will talk about will be cut from the cylindrical surface in in certain height. Okay so that's it that that's a very brief introduction of what cylindrical surface actually is. Again I wanted to introduce from this generalized concept to basically come to a very simple thing called cylinder. Just as another example of the cylindrical surface is consider this to be a polygon and these are my rulings. So invisible lines I have a dotted line right. So is this a cylindrical surface? Well the answer is yes. Our directories is some kind of a polygon and all these lines which are rulings on this surface are parallel to this one. So again we are dealing with a cylindrical surface and this thing is called a prism basically. So we will deal with certain kinds of prisms as well as certain kinds of cylinders in the future lectures and this is just the introduction to this concept. Thanks very much and good luck!