 Čutvo je trest o izbranju. Jaz stawim števi izbranje v plašku, appelj vznečati tu krušanje, imaš nekaj ali pri našega naživama. Stavim števi izbranje na konstativendi. Danes zelo. Tudi, da post kalim u vih, gledaj se o izbranje, da je viljene izbranje takv. Jebeni neko izolajte izbranje, naših, nača na stran. Let's try to build up some big families of examples. So the first type of examples I want to discuss are surfaces of revolution. Revolution actually in geometry means rotation. These are not angry surfaces. So what does it mean? It means you take a curve in some plane and let's suppose, for example, we start, and actually here I switch from our usual notation. So let's take a curve in the xz plane. Let's take a, it doesn't matter, let's take a planar curve, whatever the plane is, by changing name to the coordinates, you call them x and z. Now that means that in the x, but still I remember that there is a third one, y. So if I pick a curve, that means it lies on the plane of the blackboard in my picture, and I suppose I'm taking a curve which does not touch the z axis. For example, again, if you want to find an axis, it's enough to find a line which is not crossing this curve. Being in this plane, it means that for some choice of parameter, this curve can be parameterized in the following way. It is a function f of u, usually I call them t, the parameter on a curve, but of course this will be the first parameter in the two parameters of a surface. So now I prefer to call it u. Now it lies in the y equal to zero plane in the xz plane, so it's f zero, and then for some other function, g of u, don't be confused with the names of the coefficients of the second fundamental form. These are just two functions. It's impossible to make confusion. Now what do I want to do? On this I just assume, sorry, otherwise I have to keep on being careful about, let me call it v, the parameter on the curve, which is rotating. Now the only thing I am assuming here is that f is a positive function, just to avoid some problems. And suppose that this parameter lives in some interval, closed or open, doesn't really matter, and let's call it ab. In principle it could be the whole line r or a subinterval. Now to each point of this curve we make a rotation. We have already seen an example of this type. When we construct the torus of revolution we have taken a circle here and rotate every point. So basically now I drop the assumption into the circle and I just do it in general. So for each fixed value of the parameter v, meaning for each point here, I want to add the circle of radius, this distance, in the horizontal plane. And I want to do it for every point. That means I'm certainly going to make a disaster, but something like this. I want to build this object here. How do I do it analytically? How can I parameterize something like this? For any point, so that means how can I construct some kind of map whose image, from some domain of R2 whose image is something like this. I add another parameter, of course u, and then I will try to think where this u will live. So, such that, for example, u will be the parameter on the circle. I'm perfectly able to parameterize the circle. And actually the only thing I have to do is to put a cos. You see? This is by brute force. The image of that thing is by brute force what I drew here. Now, where does this parameter live? You see, for u equal to zero, I get the old curve f0g. For example, for u equal to pi over 2, I get another profile because this becomes zero, this becomes f, and this is still g. So that means I'm moving, when u, for example, is increasing, I'm rotating in this direction and then I get the same profile, but now in the yz plane, I mean, whatever, so the curve in this plane here. It has the same shape, but it's on a different plane. And as long as I move u, I keep on rotating this profile around. Now, if I want to cover the whole surface because the geometric I said I want to add for every point of this curve the whole circle. Am I able to do it in this way? The key point is a hole. Well, a local chart has to be defined on an open set because, of course, what is the circle? The circle would be cos u sine u with u between zero and 2 pi. But if I add the extremes of the interval, as I should, if I want geometrically the image to be the whole circle, the circle is not open. On the other hand, if I enlarge the interval, I say, well, but okay, who cares? Let's take u to be any real number. But that means I'm keeping on going around and around and around and around, many-time periodically around the z-axis. But then that would be good geometrically but not analytically because x fails to be injective and it's one of the properties of a local chart to be a one-to-one map on its image. Because sometimes the two things are fighting. I want a one-to-one map and I want it to be defined on an interval, on an open set. That's impossible. If it's open, I'm missing one point. If it's one-to-one, I'm touching one point more than once. So I'm dead. Here I have to take. I'm forced, if I want this to be a local chart, I force this to be defined on zero-to-pi, for example, open. So now our domain u, remember we always draw this picture, x and some domain u, how does it look? So this is u and this is v here is zero-to-pi and here you have an ab, whatever that is, and so basically this is defined over the rectangle, the open rectangle, and this is going to be my u in the general theory. But this is not enough to cover the whole surface. So that means that this type of surfaces must be covered by at least I have to add another map. In fact, which points are not covered? If I make this choice, the points which are not covered are exactly the ones I started with, so strange enough. I'm losing exactly this profile, but this profile is lost. So I need to add another map which covers these points, if I want to prove that this is a regular surface. How do I do it? Well, and that's the point. Now we are old enough to be able to cheat without being cheated yourself. Because I add another map which looks exactly like this, but now u does not lie here, but lies, for example, in any interval which contains zero. So basically it would be the same x, but I look at it twice, once it's defined on this rectangle, and another time is defined, and I would call it u1, if you want. And then I define it on another rectangle which covers also this, which was the part which was missing in my surface, and I call it u2. But if this is the story, I just remember it, and I stop writing it. So I will always say, I take a surface revolution and the chart like this. Then if I want to make global considerations, I know that I have to take into account this phenomenon. Now, I can also assume a curve parameterized by arc length. I mean, if it was a regular curve, there is no arm, every regular curve is parameterized by arc length. So I start with an equation, a differential equation relating f and g. Of course, the tangent vector would be f prime. Of course, when I put prime, it means the derivative with respect to the only variable, which is v. So the tangent vector would be the vector f prime, zero g prime. The square root of f prime squared plus g prime squared. And then parameterized by arc length means the norm of this, so the square root of this is equal to one. So I drop the square root and I say, I start with a couple of functions, a pair of functions, which satisfies something like this. OK? Very well. So let's see what is the curve. What are the curve at first? The curve, everything we did up to now in this specific type of example. You see, it's a very flexible type of example, because the curve is general. OK? Now, as long as, of course, so here I don't prove that x satisfies all the properties of the local chart. I leave it up to you. But here, of course, the only key point is this curve does not touch the z-axis, because those will be points where it becomes a mess, a rotation. Everywhere else, there is no problem. OK? Now, of course, you will need also this. OK? Because I never said, if I start with a curve, for example, I never said that the curve is parameterized in a way that the map is injective. But if I parameterize with arc length, I'm done. For example, in this direction, but these are all very simple considerations and now I leave them to you. Now, let's compute everything. Well, as usual, stop thinking and write. So, x u minus sine u f. And in fact, just to be quick, f and g are always functions of v. OK, I don't repeat it. So, f of v, I don't write. Cos u f g x v oh, sorry, zero. x v is cos u f prime sine u f prime g prime. Then let's go on. x u u x u u is minus cos u f minus sine u f zero. x u v x u v x u v x u v I take this and I take the derivative with respect to v. So, it's minus sine u f prime cos u f prime zero. Is cos u f double prime sine u f double prime g double prime. Then what else, n. And now, remember, these are the moments where sometimes you have to switch on the brain. Because of course I'm going to write n on one of these charts. So, nobody has told me that this surface is orientable. So, on the part covered by one chart, of course it's orientable. And I can define the usual n. So, x u, x v, divided by its norm. So, let's do it since I wrote it. I wrote them almost in line. So, this is what? This is cos u f g prime. Now, of course, the chance of making a mistake is very high. So, please double check. Sine u f g prime minus sine squared f f prime minus cos squared f f prime. So, f f prime. And then this divided by its norm. I don't even like it. I'll put them. I'll put it at the end. Sorry? Oh, minus f f prime. Yes. In fact, for once let's put what should be as a denominator. Because, of course, I switched the brain off a bit too early. You see, this is a multiple of f. So, since I'm going to normalize, this is irrelevant. I mean, I take this out because I have to take the unit normal in this direction. I don't care. I take it out. I remove it and then below there will be as a denominator there will be just what. In fact, let's write it because it's curious. It's this squared. So, cos squared g prime plus sine squared g prime. So, g prime squared plus f prime squared. But it's the only thing I know, actually. So, in fact, there is no denominator. OK? So, the parameterized by arc length here becomes nice. Well, but actually you should be able. I go on with the exercise in some sense under this assumption. But actually often we will deal with example. I mean, when I give you a curve and I rotate it, maybe this thing is not satisfied. So, remember how to adjust things. I mean, don't take the formula that we are going to get for granted for any curve. OK? In this case, we can go on in this way. So, remember this holds on the part of the surface covered by this chart. OK? Well, but actually, since we raised the problem, let's solve it immediately. So, this vector field, of course, is born on the image of X. OK? Now, the point is, does it extend differentiably to the points which are not covered? So, in this specific case I know that X touches the whole surface minus this curve. So, at every point I have a normal vector. OK? At every point minus this curve. So, I can ask, if I go to the limit here, is this an extendable to a differentiable object? And I can see it immediately. Of course, this is not a general vector. That's not what you do in general. Because, of course, if you have a surface which is covered by many charts extending from one chart to the other it's OK. But, I mean, here the point is that the whole surface is covered by two and the points which are missing in one are a closed subset. OK? So, what does it mean? The limit as u tends to 0 or u tends to 2 pi. OK? On one side or the other. Is this vector field extendable in a differentiable way? Yes. It's already written here which is the extension. OK? This formula, of course, makes perfectly sense when u is defined over the whole R. OK? So, this one extends and this is orientable. OK? Now, we probably have everything to start computing first and second fundamental form. What is e? It's sine squared f squared plus cos squared f squared. So, it's f squared. Now, don't be confused. G is what? It's a scalar product between these two. So, minus sine cos ff prime plus sine cos ff prime so G is 0. Sorry, capital F. OK? And then capital G. Let me write it here, so I save a bit of space. It's cos squared f prime squared sine squared. So, it's f prime squared plus G prime squared. So, again, it's 1. But here, please, remember, the curve is not parameterized by arc length. If you use this formula, it's a disaster. If it's parameterized by arc length, it's 1. And now, little letters. X u u n. So, that's minus cos squared fG prime minus sine squared fG prime. So, it's fG prime plus 0. So, it's minus fG prime. OK? Little f in the sense of the second fundamental form it's X u v n. So, it's minus sine cos f prime G prime plus sine cos f prime G prime plus 0. So, f is equal to 0 and little g is this one times this. F double prime G prime plus sine squared. So, it's f double prime G prime minus f prime G double prime. OK? So, if I ask you which are the principal curvatures be careful because now this looks like already a diagonal matrix. The coefficients of the second fundamental form. So, what we have to do remember in the in our notation of the proof is to compute the matrix A. OK? So, since this one will come in as a renormalization so the matrix once I know this which are simple to compute I compute the matrix associated to minus dn. So, we have to the standard basis to remember eG sorry eF fG to the minus 1 eF fG OK? So, then I compute it in my particular case I substitute here what I found and this becomes f squared 0 0 1 inverse times minus fG prime 0 0 and this expression of second order f double prime G prime minus f prime G double prime OK? Now the inverse of this I don't even have to think it's 1 over f squared 1 0 0 So, basically this is dividing by f squared here so this becomes the matrix minus G prime over f 0 0 and this expression f double prime G prime minus f prime G double prime divided by f prime divided by f squared OK? Now, you can say that these are the principal curvature so you see there is a scaling factor which comes in so the difference between the eigenvalues of this square is this object here being diagonal it means just it's renormalizing OK? Now, I don't really know which one is smaller than the other depending on the functions f and G OK? Now, because I should say now k1 is equal to this and k2 is equal to that or maybe it's the other way round I don't know but these two are k1 and k2 at the corresponding points and the Gauss curvature is the product of these two OK? And the mean curvature is what it should OK? Now, of course, this is different from what I got in my notes so where did I make this experiment? So, ten seconds to find either a mistake here or a mistake there OK? As a sub-exercise is to double-check if everything is correct now I don't have time to find what? That's already a good... Sorry, you are right OK, OK, OK divided by f squared everything No, this is dividing by f squared only this one. OK, very good So, the exercise is done and these are the two principal curvatures Now, the only thing you can argue is if... Do we know anything about this function here? Well, not much, but certainly can be simplified a bit so just let me double-check if now OK, so now we are in perfect shape Now, is this function something that we can simplify in some sense, yes remember you have this equation here OK, so I can take the derivative of this and I get an extra, a new equation of second order because here there are second derivatives coming in and which of course are which is what? I mean, of course here I would get twice, twice and zero so the two I drop it immediately so this becomes f prime f double prime so that equation if I differentiate it, I get f prime plus f double prime plus g prime g double prime is equal to zero OK but that means so, for example, the Gauss curvature which was the product, so this implies so k was independently of this equation was just the product of these two, no? So it was here there is nothing interesting to be discovered f double prime g prime minus f prime g double prime but here now I can make a substitution and here, OK, I leave it to you it's just f double prime over f you see, well it's just taking here you have g prime g double prime you take it out of this equation you put it back and you see what's going on OK now because, of course, this is a simple expression that you can analyze so, for example, when what does it mean, it's positive or negative so, right, we need to well, but independently I mean, remember the geometric interpretation we gave that's why we wrote the geometric interpretation of elliptic points and hyperbolic points in words to avoid exactly so the interpretation by saying if I cut with the tangent plane the surface lies on one side or it crosses the tangent plane this is a way to say independently of the choice of n OK that's why instead of saying the formula we wrote all this dissertation in English OK so, for example, it's clear that this point should be an elliptic point and, in fact, all the points rotating rotating, you see, in fact first observation, this function now, let's go back to this depends only on v the Gauss curvature depends only on v and not on u which it's a good check it had to be like this OK, meaning if some point has some property then every point on the circle has the same property points on the same circle are, of course, absolutely the same OK, so if something holds here it has to hold over the whole circle and this point should be clearly elliptic OK why, for example, this point should be hyperbolic in one direction of rotation which turns left and one direction of the curve which turns right meaning if I cut with a tangent plane I cross it OK and so on and where it's zero F is a positive function in any case it's a denominator but I mean this is positive so the only way this is zero is where F double prime is zero and so on the other direction here are zero OK that's it that's a nice family of examples you see the remark I made about the orientability of this type of surface you can now play use it also here this formula hold on the part of the surface which is covered by this map X where X is defined with this bounce so now if I ask you how much is this Gauss curvature here in principle you should not be answering you should shut up and repeat the computation with the other chart but being the chart given by the same function just defined on a different interval the final formula will be the same so in fact this formula holds globally this is an accident of this specific situation of course there is nothing general here it's a strange type of surfaces which are not covered by one chart but by many charts for example two but somehow the analytic expression is the same so whatever I compute holds globally very well now just as a specific example in fact you know one example of this of course well the torus we did it you have a question? but you need to speak much louder I mean I am used to two years old kids so the average tone of the voice is above shouting right but I remember mentioning to you this problem when we made the computation in fact one of you came in my office and the question is there is a discrepancy between this and what we did in class when we proved this formula for representing first, second fundamental form and the matrix associated with this now there was a problem there there was a minus one missing or for some funny reason you switch N because of course you have this freedom and you decide that instead of N you use minus N so if you go back there and you try to be a bit more precise than what I was you should, if I remember there is a minus there missing right now specific case the torus of revolution so you can apply the sphere is of course another example let the whole sphere with this trick because you want a curve which does not touch the axis of revolution so if you want to draw the sphere as a surface of revolution you are bound to take this curve of course the profile of half a circle but without the north pole and the south pole and now this problem of the north pole and south pole and we faced here it's another problem so if I take another chart of this form I will never cover the north pole and the south pole ok so the sphere minus two antipodal points you can put it into this picture if you want to curve of course you can play another game just for the accident that the sphere so many symmetries of course the sphere is also the surface of revolution ok so in some sense it is a surface globally you can define it as a surface of revolution or you do another in fact more interesting trick is that you analyze what happens to this picture when the curve touches the axis of revolution because really what we feared is this kind of situation for example if you rotate something like this certainly you are not getting a regular surface this is kind of casps ok these points are clearly bad but so this shows you that there is a problem even if of course there is a bigger problem if the curve crosses the axis but even if it doesn't cross there is a problem but if the curve touches the axis of revolution with a velocity orthogonal then the problem disappears at least at first order so here the problem is smoothness ok so c1 you just require that this is orthogonal if you want more you require more conditions ok so it's not that if the curve touches you are dead there are extra conditions in the case of this sphere all these extra conditions are satisfied but if you rotate this profile you still get a regular surface ok anyway another specific example famous specific example of surface of revolution since it's important for another special lecture I would like to give you later I guess in fact Thursday probably well we have seen for example the elliptic paraboloid for example of surface of revolution one not every elliptic paraboloid because elliptic paraboloids could have ellipses as sections the one we studied in a previous example had exactly a circle as section in that case it's a surface of revolution ok now there is a very famous surface in this family which is called the catenoid oh what is the catenoid the catenoid is what you get by rotating so remember our choices were the two functions f and g little f and little g which gives the curve and then everything becomes automatic so in this case I write you directly the map x of v there is no point in putting another parameter times cos u hyperbolic cosine of v sin u u v a v ok so basically this a is totally relevant it's just one parameter a shrinking parameter so suppose it's just a positive number you see in this case I've chosen g to be this function and f is a cos cos ok how does it look in the so you see in particular besides dividing by a which is not important you can think of it as a graph of a function so forget now the u so meaning we are looking at the curve a cos v 0 a v so you see the x component is a graph over the z component ok so if I draw it if this is the z axis I think of it and this is the x axis I think of it as a graph in this direction ok it's a graph over z which gives me some x and what is the function which I should draw cos ok but cos I know it's something like this actually here parameters are so v is any real number I should always be careful about so where everything is defined and u is between 0 and 2 pi for one chart minus pi pi for another chart ok so if I draw the whole thing I get something which looks like a parabola in fact this curve if you put it in the standard picture as a famous name it's called a catenary so the curve itself in fact the curve itself was very famous you know why have you ever discovered this curve in bridge sorry in bridge meaning you have even studied some kind of static or dynamics of buildings that's the place where it was born in fact it looks like especially if you put it the other way around upside down it looks like an arc so that's exactly the shape of an arc but it's also in fact that's kind of second step the first step when it was born it was born in late 17th century to solve one of the most simplest problem actually so the calculus of variations I mean mathematical physics call it how you want it was born by looking at very simple problem if you look back still enough to create differentiable calculus pick two points and pick a wire is this long enough I don't want to take my belt as a wire otherwise this lecture becomes too informal pick a wire a long wire and take two points hold it at two points at the same height now there is gravity there is a force in nature and there is a resistance on the wire along the wire there is a resistance because of course it has some kind of tension so which is the shape of the wire you know mathematics or mathematical physics was born through letters through challenges in the old times they didn't have matzinet or email or publishing journals so the way think somebody made a discovery and wrote to everybody famous in mathematics saying I am the best mathematician alive because I know how to solve this problem are you able to solve this problem too actually this happened historically many time many important discoveries were actually done in this way because of course if nobody could in a reasonable amount of time answer back that really meant that this guy was one of the best and so some king some prince son I would hire him or her well at those time her was very rare but was hire him and cover him with gold because of course for a prince or a king to have the best mathematician alive at those time it was important now nobody cares so so what happened with this curve so this was the challenge that Galileo made to everybody I can be slightly me yesterday evening I was not able to double check what I'm saying but I'm not sure if Galileo was the one answering or the one challenging take a nice book of history of mathematics and you will find the real truth but if I'm wrong I'm slightly wrong and he said oh I know the solution in fact he wrote it in a book and the solution is the parabola Galileo was already Galileo Galileo is the best the parabola for many years this was considered the solution and then somebody came up and said ok Galileo is Galileo is very good relativity Jupiter whatever you want but I keep on finding something else not the parabola in fact I find the catenary which the name is really catenary in Latin means the shape of a wire ok so then discussions no you are wrong no Galileo this is the truth the catenary was born in this way so it is one of the most famous by Galileo now at those time they didn't know what hyperbolic cosine was ok it was one of these we will see in 20 minutes another famous example of this phenomenon ok but now you have everything now I erase the formula but now they are easy because you see g is essentially the identity times a so g prime is a g double prime is 0 f is this so f prime is a sinh so you don't switch sine and double prime is you put everything together so I don't repeat the computation and in fact I tell you also the mean curvature which I didn't write because it's that long formula all the letters e f and g capital and lower and h turns out to be 0 identically 0 k turns out to be minus in fact minus it's ok because this is kind of a convex function so every point you expect every point to be hyperbolic because you see at every point the tangent plane will have one direction going one way and one direction going in the opposite way so it will cross the surface at every point so every point should be hyperbolic so it should be minus something and the something is simple it's just a squared kosh squared kosh 4 v this is just by substituting substituting substituting with care because in this case this is not a principally a parameterization by arc length of the catenary f prime squared plus g prime squared is not one so you have to take the formula corrected with the right denominators remember there was this denominator at some point was one and now it's not one ok so another example another famous example again this was born in the same way this time was not Galileo to make the mistake you take another famous curve arising in nature the problem is this one it has many strange famous name but basically suppose there is a donkey here connected with a wire to a stone here ok so this is a donkey in mathematics and now the donkey starts moving or the ox depends which animal you prefer starts moving in this way at uniform speed so that's a very standard situation you have something on your back but you move orthogonally you start moving orthogonally and this is the trajectory of this object ok well you expect it to be something like this ok more or less well let's put names to the more or less and I take this curve here and now instead of doing I rotate it in this direction actually it's a nice exercising calculus so describe the differential equation so this object this curve will be described by two functions F and G it's a curve in the plane so which kind of differential equation does it have to satisfy to be the solution to this problem of course here the constraint is that this wire has no fixed norm I mean it's a piece of wood it's not a spring it's not elastic ok so I can tell you directly the end of the story so X of UV in this case actually by changing names to the coordinates I will keep on using the same notation of course it's enough to call now this Z and this X ok I don't care and this becomes Kosh AV sine U and then G I should put here G of V ok but G of V I gave you just implicitly the definition so this is the integral between 0 and V of 1 minus A squared sine hyperbolic sine squared of AT everything to the power one half and don't be surprised that this looks like the norm of a vector just the way it arises dt ok now in this case this curve is parameterized by arc length I mean this is the convenience of this in some sense you do you it's more difficult to draw but analytically simple because G prime of course is this function if I take f prime is this f is this so f prime squared plus G prime squared so I throw away this half and I get one by the standard identities of hyperbolic functions ok Kosh minus sine is equal 1 ok so here on this example you can really apply the formula without being careful of the denominators I was mentioning in the case of the catanoid and this becomes interestingly enough very interesting thing is the gauss curvature also in this case you are expected to be negative just by looking at the picture ok but actually the little surprise is that it's a constant ok it's a surface of negative constant curvature ok of course this is non-compact so it's not contradicting anything in fact you might wonder whether there is a theorem saying in the non-compact case that this is the only possibility now this actually I forgot to tell you the name of the curve the name of the curve is the tractrix this is of the curve in fact the surface I don't think it has a famous name by itself it's the revolution of the tractrix ok of course Italians call it the dinis surface but I suspect it's just nationalistic I mean the implicit function theorem in Italy is dinis theorem so nobody knows outside Italy I suspect that even in this case this is not the official name please there was a question no anyway if you have a favorite book of history of mathematics this was the curve not the surface the curve was at the center of another of this discussion ok in this case I think it was Newton solving the problem I think Newton was challenged by somebody that I don't remember and he came up with the explicit parameterization of this curve now ok so this is enough as general theory and specific examples of surfaces of revolution in the last half an hour I'd like to talk to you about the other famous big family of surfaces which will help us making geometric considerations about the Gauss curvature and these are called ruled surfaces in some sense you would like to say that the ruled surface is a surface which has the property that for every point there is a straight line passing through this point contained on the surface it's covered by lines by straight lines ok so how do we get kind of a more formal definition of this fact so first let me define what is a one parameter family so one parameter this is just don't make a big effort to remember these things these are just to speed up later one parameter family of straight lines is a pair alpha of t w of t is a correspondence let me write everything even though it's quite clear it's a correspondence that assigns to each time t in some interval i ok a point in some sense a map which defines for any t a point alpha of t in r3 and the vector w of t the only thing in trivialities I want this vector to be always non-zero otherwise everything becomes really stupid ok and of course so basically these two in some sense alpha and w are vector fields but in some sense I'm thinking of alpha as a point and w as a vector ok there are two vectors in r3 and of course in a differentiable we have to be differentiable way ok so what do I do with something like this if I have a one parameter family of lines of course that means so I'm thinking so you see if I want to draw I should draw basically two curves in r3 in some sense I want them to interact in the following way for each time t I want to think at alpha as kind of the base point a passing point and w as the directional vector of a straight line ok so I would really like to think of alpha as the base point and w a vector pointed here and I want to consider the whole line through alpha with this direction ok this is the way I want to use alpha and w well what does it mean that if I have something like this I look at so what is this curve this line what are the points covered by this line well for any t so now this will be x will be a function of t and v I look at v w of t so that's the way so for each t if I let v varying on r I'm parameterizing exactly this line ok so now v is any real number so t varies where it should but v I can take it any real number so I take the whole straight line so the image is not like this so the image is called a ruled surface because really has a set it has the property I was telling you at the beginning so as a set the image of this map has the property that there is a point and the vector in fact for every point of the image of this do you have a little geometric intuition when alpha is moving in some way in time also w is moving in time so w for the same t was here when alpha is here maybe w is here and when alpha is here maybe w is here and so that means I'm keeping on adding these lines so this object as a set this is the property I was telling you because now give me any point of this set it has to be on one of these lines so ok that's it give me a set and so that means through this point there is one straight line all contained in this set by definition now these lines so let me give a name even though I don't think I'm going to use it so for each t I have a special line this really so it is the line alpha of t plus v w of t meaning now I think of t as fixed and v moving is called so this line the line is called or the family of this line is called a ruling of this surface alpha and that's it we don't give a name to w in some sense it's the family of directions of the rulings so examples because now we have to be much more careful than before at the first step well of course example 0 so do you know rule surface which rule surfaces do you know that's already example 2 probably for example 0 is the plane but that's already very interesting now it's 0 plus it's 0.1 because suppose it's exactly the plane of the blackboard ok why do you tell me that this is a ruled surface well you have to produce me alpha w and tell me that the plane is the image of an axle you said 2 axes what does it mean 2 axes that's what I heard if you tell me something obvious I have to say well probably ok but what is alpha when you define it as 2 axes what is alpha x and w so so first proposal use cartesian coordinates in that case we tend to say alpha of t is something like t0 I guess ok so the point on the x axes now use the other coordinate as direction so that means w of t is what it's constant it's 0.1 now it is true that every point of course is of the plane is of the form alpha plus v x t equal to x v equal to y ok, one possibility by the way I saw hand waving showing me circles or in fact somebody doing like this which was the most mysterious to me you are not looking at me anymore but you did something like that I don't understand what does it mean well in some sense for example polar coordinates is another obvious choice alpha of t is equal to a point for example in a unit circle but you don't care of any radius cos t sin t now I'm thinking parametra is the plane using in some sense polar coordinates that means I move it here and of course every time I'm here I take the line in the same direction so in this case w of t is again cos t sin t it's the point itself 0 0 as alpha of t sorry, what does it mean oh, that's another interesting proposal alpha of t is equal to 0 0 so your colleague is saying why move the point stay still in the origin and move the vector in the same way and move the vector again in fact this is even more polar coordinates in some sense the plane was 0 0.0 plus the example but it's already telling you that maybe there is absolutely no canonical way in fact why one should be better than the others no, there is nothing better than the other here but there is no canonical way to given a set which has the geometric property to actually write down alpha in w so this was already a good so do you know other examples? so in the case of the cylinder and I think by the cylinder you mean the one constructed over the circle and then you add vertical lines ok, now we are immediately going to generalize this but in this case why this should be a ruled surface in this language again you can pick alpha of t to be the circle and now w of t is always the vertical one sorry, now it's a space curve so for example in the z equal to 0 plane and this is 0 0 1 at every point of this I add the vertical line so this is of the same length in fact let me give you a general definition of cylinder now because now it's clear that the circle has nothing interesting in fact suppose you have a planar curve alpha of t suppose alpha of t is contained in some p so you start with a curve in a plane wherever it is w of t is constant so what's really going on here now the plane might be anything you have a curve in a plane so maybe this plane will have a normal vector but I'm not even taking the normal direction so you see how many accidents there are in this example I'm taking the circle and the normal to the plane why I build a general cylinder there would be something for which there is a curve in a plane and one direction maybe but the point is at every point of this curve you need to take the same direction so even on the circle why taking the orthogonal cylinder somehow you can take something like this and we still call it a cylinder we call and direction whatever the vector w so this would be a general cylinder but then of course there is another obvious family of ruled surfaces which are cones in some sense the plane is everything the plane is a cylinder it's a pathological cylinder if you want in this picture here you see that it's a cylinder it could be constant and outside the plane of course that's a very pathological situation but why not cones cones are what how do I define in this language cones so it's a ruled surface where again alpha of t is contained in some plane p which remember is not required by the general definition of ruled surface so you start with a special curve mainly a planar curve and the rulings all pass through a given point p0 so what does it mean not now well ok there would be the kind of pathological situation but let's try to avoid a little bit pathologies too much ok so of course what is this point this point is the vertex of the cone but you see again I'm not drawing I mean it's a clear generalization of the standard picture because I mean of course the standard picture is this ok so alpha would be for example one of these sections for example and w would be the vector which makes it going to the vertex at every point or you can be even smarter you can take this section alpha is constant and w is rotating in this circle here that defines the same object ok that's another way to put it very well so these are kind of general families so cones and cylinders let's see the most famous ruled surface besides this which is one of the quadrics so it's something defined by a degree 2 polynomial in R3 so we are at example 3 example 3 ok example 3 hyperboloid of revolution one sheet hyperboloid remember to pronounce it with a long e ok now what is the surface s so it's something you want it has to be of revolution so there has to be remember the trick of classifying quadrics is by cutting with planes and seeing which kind of conic first you classify conics then you pass to quadrics and you say ok so it has to be of revolution so one of these sections must be a circle otherwise there is nothing rotating hyperboloid of one sheet means the other sections another of this section is an hyperbola and the third section one two and three it's another hyperbola so two hyperbola and the circle that means the equation is like this x squared plus y squared minus z squared is equal to one geometrikally well geometrikally it looks like this I should have first drawn something like this this is of course a surface which is very popular in Italy because we tend to see it in fact in Italy it has an Italian name it's called the spaghetti shape now because one minute of cooking class how do you cook spaghetti properly first you boil the water first without putting the spaghetti inside ok, don't do that you boil the water you take a bunch of spaghetti you put them as a cylinder that's easy now if you are not Italian you throw them in the water and then they get stuck and you throw them away if you are Italian you put them in a cylinder hands in this way and you twist one way one direction up and one direction below and then you put them in the water and they don't become they don't glue now or anglosaxons which are clearly much less fantasy people they have a very sad life ok because that's the typical shape of a being of the treasure in your office ok, having said that now the spaghetti picture besides having one minute of relax was good to see the two rulings so this surface it's a regular surface you prove it by regular value inverse image of regular value I don't know you are old enough the two spaghetti the spaghetti gets in a position where you can see that there is one family of lines somehow going around in one direction and at the same time so this is special now besides jokes because it has two independent rulings ok, you see this line belongs to a family which really rotates goes around and when you are back here really it's again this one and this spaghetti when you rotate it goes around in a line because of course spaghetti are not flexible and gets back here to the same line so that means you can write it down independently as in two ways with the same alpha of course I'm taking alpha to be this section one of this section ok how do I do that now let's do mathematically this picture well, a simple way it's to parametrize this surface in this form it's cos s minus v sin s this is one of the two rulings actually sin s plus v cos s v ok ok, it's simple that this satisfies this equation and then you have to prove also the other way around that for every point here there is an smv for which is of this form if I take this squared plus this squared minus this squared it's automatic if I haven't done mistakes it's automatic so of course I can split now once I write it in this form sin s zero and this actually will be my alpha in the z equal to zero plane plus v something which will be my w and in this case is minus sin s cos s one ok so this is one of the two in fact the other one is just by switching ok, so this is alpha and this is w now what is special about this now of course s has become t well an interesting thing is of course that w, if you look at it is really the derivative of alpha plus a constant so here there is the accident here is that w is alpha prime plus the vector zero zero one now if I didn't add this it would be funny, you have now suppose we are in the plane and we don't like what otherwise the picture becomes too messy ok, I'm taking a circle because alpha is a circle if I take w to be alpha prime without this of course this is still a vector in the same plane and what vector is it it is the tangent vector remember before we ok, so and of course that's why I'm saying switch and so on now of course if I think of the surface in principle I could use this alpha in this w and what do I get a disaster which actually makes me because which object would be the image of alpha plus v w the whole plane minus a disk I mean I cannot hit any point inside the disk in fact in more than one way in general ok so really the image of the map here would be not even a surface so big warning because now we are unfortunately it has become too late and now we are going to make the theory I mean our old theory applied to this case but here the problem starts immediately the image of X is often not a regular surface for any sorts of reasons one is this kind of monster for example but another one is that clearly where it's written that X is injective remember X of TV if I really hope that this is a local parameterization at least for a piece I mean it's alpha plus t plus v w alpha of t plus v w of t but how can I hope that this is injective so why this should not be equal to alpha of s plus u w of s for some choices of tv and su but this is in general going to be completely false think of the cone for example it's already here the cone is the prototype situation where if every line passes through the same point this map is orthogonal to be injective for example so we have to do something but this something will come on Thursday now questions the other thing which other thing sorry either you start speaking very loud I'm going to ignore all the questions is it wrong? let's double check it x squared so that's cos squared plus v squared sin squared minus twice the minus twice I throw it away immediately the minus twice goes away so this is cos squared plus sin squared so this is the one and I throw it away here so let's see if everything else disappears everything else is here is what? it's v plus v squared sin squared plus v squared cos squared so it's plus v squared minus v squared how do I prove the other inclusion because actually this is a twin if I want to say that s is equal to the image of x of course what I just checked is that the image of x lies here so the objection of your colleague is how do I prove the other way in general it's actually tough but now fortunately one of these coordinates is already z ok if one of these coordinates I mean if you know I mean this gives you the hint so call z so this object is this one so I call v z or by brute force so if you give me this in x, y, z I need to tell you I need to find s and v for which ok so how do I get v because I take the z and I call it v now the problem is how do I find s ok but now it's easy again because give me a point here what is s s basically is projected down here you have a circle and call s if you want the angle or minus the angle or pi minus the angle whatever you want but essentially the angle of this point with respect to the direction if you want this to be cos s s will be the angle with respect to the x axis because for s equal to zero I want to find x the x axis so s will be the angle with respect to the x axis so now I define you v and s and this by construction so now I have to check that with my choice of v and s x of sv is equal to exactly to the point you gave me are you with me? the feeling I lost you so you ask me give me a point x0, y0, z0 I need to prove you that this is x of sum s0, v0 ok of course this on s how do I do it? I tell you v0 by definition I define v0 to be z0 I define s0 to be the angle between the point x0, y0 in the z equal to zero plane so I take the point that you gave me I put it down here and I look at the angle with respect to the x axis and I call it s0 now I know it's true but now I have to check that this is s0 and this v0 this is true but this has to be true I mean now by the way I construct that's not v0