 Hi, I'm Zor. Welcome to a new Zor education. I would like to have a couple of words about similarity of objects, which are not the ones which contain just straight lines and planes. So it's not like cubes or pyramids, for instance, a little bit more complex. What I mean is cylinders and cones and spheres. Well, similarity in two-dimensional space on a plane, mostly about straight objects, straight lines and angles. Similarity of non-straight objects is a little bit less of a concern because the only object which consists of non-straight lines on a plane is basically a circle. And all circles are similar to each other. So there is not much of a theory behind it. And obviously all circles are similar to each other because similarity is based on scaling and scaling preserves equality. So if two points of a circle were on the same distance from the center and you scale this circle, then two corresponding points will also be an equal distance from the corresponding image of a center. So it's kind of simple. In three-dimensional space, as I was saying, there are a little bit more complex figures which contain non-straight elements. And I would like to spend a very short time to basically talk about this type of similarity. Well, I do suggest you to watch this lecture from munisor.com. It contains whatever I'm talking right now as a text as well. So there is a video presentation and there is a text to it which you can read as a textbook. All right, so the first one is cylindrical surface. Okay, first of all, let's recall that we did discuss similarity in three-dimensional space as a scaling and scaling preserves the straight lines. So the image of a straight line is a straight line. It also preserves equality between the lengths. So if two segments, for instance, are equal to each other in lengths, then after scaling the resulting segments will be equal. So the length is preserved, the equality between the lengths is preserved and angles are preserved in the scaling. The image has exactly the same angle as the source because the image of the straight line is always parallel to the source. If you have a straight line and this is some kind of scaling with this center and whatever the factor is, the resulting will be straight line parallel to the original one. And I'm talking only about three-dimensional space right now. All right, so angles are preserved. And by the way, not only angles but also dihedral angles are preserved. These are all topics we did discuss before. So right now we're talking about cylindrical surface. That's my first topic. Now let's just recall what is a cylindrical surface. You have a generatrix which is a straight line and you have some kind of a curve which is called directories. And now through each point on this directories, we draw a line parallel to our generatrix. So the shape it might actually take. So this is a cylindrical surface. Or you might say it's the result of a movement, if you can introduce this dynamism, movement of a straight line which is drawn through one point parallel to the generatrix as it moves parallel to itself along the curve. It doesn't really matter what kind of approach you use. It's basically the same thing. Now what if we take some kind of a point as a center of scaling and scale the whole picture. So every point of this curve will be somehow transformed into another curve, right? But now let's think about it. Since image of the straight line is a straight line parallel to original, so the image of the generatrix will be another generatrix which is also a straight line parallel to this one. And each of these will be parallel to the new generatrix. Which means it will be actually parallel to itself. So this parallelism is preserved and that's why cylindrical surface is transformed by scaling into cylindrical surface. So that basically is all about cylindrical surfaces. Now before going to a circular cylinder we have to just mention something which is quite obvious but still needs to be discussed. What is the image of a circle? Now I did mention that on the plane image of a circle is a circle which is similar to the original one. Now why image of a circle is a circle? Well, because the distance is preserved. So these two points were on the same distance from this. Now you scale the whole thing and you have a bigger circle as an image. Now why is this a circle? Because image of this point will be somewhere here, image of this will be somewhere there and these distances will always be equal to each other. So if this is set of all points which are equidistant from the center and the distance is such and such then we scale the radius and it's bigger in this particular case but these also will be all these points which are on the same distance because equality between lengths of the two segments is preserved. Now, same thing absolutely analogously is in the three-dimensional space. So it doesn't really matter whether it's a plane or it's a three-dimensional space scaling preserves the circularity, if you wish. So if object is circular which means it contains all the points which are lying in the same plane which are on the same distance from the center of the circle then the base plane will be converted into a plane by scaling and since the quality of the lengths of the segments is preserved then the image would be a circle as well. So a circle is transformed into a circle. Okay, great. So what we have right now the cylindrical surface is transformed into a cylindrical surface but now when we're talking about a circular cylinder it's a cylinder where the director is a circle in some plane, right? Now, I didn't draw a circle which looks like a circle because it might be in the space so I might just look at this at the angle. So this is a director which is a circle in some plane and now we have all these lines parallel to this one and that will be our circular cylinder. Now, why the image will be circular cylinder? Well, number one because a circle will be transferred into a circle also on the plane, on another plane parallel to this one by the way it may be bigger, it may be smaller but it will be a cylinder, I mean it will be a circle and straight lines which are parallel to directors will also be parallel to the new image of the directors so the image of this circular cylinder would be a circular cylinder Now the surface, this is the side surface, this is the base now similarly there is another base here which is parallel to this one and the parallelism is preserved so again the image would be a circular cylinder and finally we are talking about straight circular cylinder that's when these lines are perpendicular to the base so this is a director, this is a generator which is perpendicular to the plane where the director is located and now we have all these lines which constitute a cylinder Now, why is the right circular cylinder preserves this type when you are scaling? Well, because the perpendicularity will also be preserved what does it mean that the line is perpendicular to this base? Well, it basically means that at least two lines on this base are perpendicular to the generator and all these lines which form the side surface so this is perpendicular to at least two straight lines on the base plane and again when we scale the whole thing the base plane will be transformed into another base plane and these lines which form the side cylinder will also be parallel the circle into a circle and right angles will be transformed into right angles so the perpendicularity of the line and the plane is preserved when we are scaling everything so the line will be transformed to another line parallel to this one the plane will be transformed to a plane parallel to this one and this perpendicularity is preserved OK, so right circular cylinder is transformed into right circular cylinder so that's why we can talk about similarity in this particular case Now, I did not say, and that would be incorrect if I did I did not say that all right circular cylinders are similar to each other because for instance one can be thin and another can be short and fat and they are not similar to each other because similarity assumes the proportionality and this one is supposed to relate to this as let's say the radius to the radius and in this particular picture that's not the case so I'm not saying that all the cylinders are similar to each other but if they are similar then one of them is being right circular cylinder then another is also right circular cylinder so from similarity follows the identity of the type OK, what's our next figure? cone OK, now remember what cone is? you have an apex which is a point and again some kind of a curve and we connect each point of the curve with an apex and that's our cylindrical surface well again obviously scaling preserve the cylindrical surface so the similarity preserves the conical type of the surface because the straight line goes into straight line now, what's interesting is a circular cone now the circular cone is the cone when this particular directress is a circle now again let's just use whatever we were talking before if we transform it using the scaling the point will be converted into point a circle will be transformed into a circle and these lines which connect point to point will be converted into some kind of lines again straight line so the image of the circular cone is a circular cone in the operation of scaling OK, now the next one is the right circular cone now what is the right circular cone? that's the one when apex perpendicular is dropped from it onto the base goes to the center of the base of the circle so question is during the operation of scaling would the right circular cone be transformed into right circular cone? I mean we know it will be circular it will be cone and it will be circular cone question is the right is transformed into right circular cone well yes, obviously the reason is yes and again the point is that the angles are always preserved so if you connect this with the center of the circle now it's a perpendicular right? now the image of this would be another circle and since these angles are preserved and these are right angles so this would be right angles so the line connecting the apex to the center of the circle would be perpendicular to the whole base plane of the transformed image center goes to center circle goes to the circle maybe a bigger radius and the point to point so this line would be transferred to this line and since the angles are preserved it would be perpendicular so the right conical circular surface right cone, right circular cone is transformed by scaling into right cone again question is whether all right circular cones are similar to each other the answer is no and again the counter example is exactly like I was saying about cylinders if one cone is a narrow one and another is the wide one so the proportionality between let's say the edge and the radius is not the same as here and the proportionality must be preserved by the scaling so not two cones are similar to each other but if one is the result of a scaling of another then this is the similarity and the similarity preserves the type right circular cone will be transformed into right circular cone my last geometrical object is a sphere and sphere is probably the easiest among them because what is a sphere? this is the set of all points in three dimensional space which are on the same distance from a point which is called a center now the distance would be different in the image but equality between the distances so if these two points have the same distance from the center their images will also have the same points from the center so basically all these points which constitute a sphere will be transformed into some kind of a surface which contains points which are on the same distance from the center among themselves which is a sphere now there is still maybe a slightly open question whether this is all the points which are maybe there is some kind of hole here which does not have any prototypes not no source but that's not the case because you know you remember the scaling is a symmetrical transformation so if I'm scaling from this point with the factor of f from this to this and using the same point with the factor of 1 over f I will get from there to here so all these kind of considerations that there might be holes in the image that's not really true because there will be holes in the source in the reverse transformation so let's not go into these very delicate points because there are some underwater currents in this logic but intuitively it's obvious and that's enough, completely enough for this particular lecture that if these are all points on certain distance from the center these are also all points on the distance from the center which is related to this distance with some kind of factor f all right, but now and this is the only case when I can say that all spheres are similar to each other and this is actually very easy because there is only one dimension which defines the sphere you see, in cylinder case you have two dimensions you have a radius of the base and you have the height or altitude and these are completely independent dimension and since they are completely independent of each other they might be in different cylinders they might be different so the proportion can be different between this and this and one cylinder and this and this and another cylinder we have two different parameters and they can be non-proportional to each other that's why I cannot say that any two cylinders are similar to each other same thing with cones also two parameters there is a radius and there is a height two parameters are independent of each other with any radius and any height I can build a cone so I cannot say that any two cones are similar to each other because the proportionality might not be preserved but in case of a sphere I have only one parameter which defines the sphere which is its radius and since it's one parameter then any other sphere is similar to this one with the coefficient of proportionality equals to the ratio between the radiuses that would be my factor by which I have to scale and how can I find for instance the center if I have two spheres in the space can I find the center of scaling well actually yes I mean if you think about it's supposed to be on the same since center goes to center that means it will go to so the center of scaling should be on the center line between the spheres now how can I get the how can I get the point on this line which is actually a center of scaling well there are different ways but for instance you can think about plane which you just position in such a way that it touches two spheres and wherever this plane intersects this center line that would be the point but probably we will have to discuss it in a little bit more details when we will talk about spheres and planes which are tangential to sphere etc alright so basically that's it for today I suggest you to go again into unizord.com and look through the notes for this lecture and other than that that's it thank you very much and good luck