 Last time we talked about the simplest curves and surfaces in space also known as lines and planes and Today we'll talk about more more complicated objects of the same kind also curves and surfaces Okay, so let me just briefly remind you what we did last time. I know it's a warm September afternoon and And you want to be elsewhere some of you maybe but you will you will get there just You know the sooner we start the sooner we finish. Okay, so I Want to use from now on I will be using this notation R3 for the three-dimensional space so R is for real numbers and then we write R to the n where for the n-dimensional space So R2 would be the plane and R3 will be the three-dimensional space Okay, so oftentimes it will be convenient to just abbreviate instead of saying space Now lines and planes in R3 Can be can be defined or represented in a very concrete way which we discussed last time for lines It's a parametric form which means that each of the three coordinates in R3 is written as a function of an auxiliary Coordinate which we call t, but it could be any other coordinate any any other letter you like So let's use t so then it the formulas will be look as follows axis x0 plus a t and y is y0 plus bt z is z0 plus ct where These numbers are given this are x0 y0 and z0 and a bc are given and they correspond to the following x0 y0 and z0 are coordinates of a particular point on this on this line and abc is A particular vector which goes along this line we call it direction vector So that's a bc So each of the coordinates is written as a function of t and this tells us Parameter is a very gives us an explicit parametrization of this line in other words for each value of t Each value of t gives rise to a particular point on This line for instance t equals 0 corresponds to this point T equal 1 corresponds to this point and so on now for planes We did something different for planes. We did something different a plane doesn't have a direction vector Right a plane is not determined by one direction is determined by two directions or two vectors like like so That's why it's a two-dimensional object if it could be determined by one vector. It would be one dimensional like a line so in fact if we wanted to imitate the same procedure for Planes we would have to choose two independent coordinates We cannot parameterize the entire plane which is a two-dimensional object by one coordinate. We have to parameterize by two coordinates Now it's possible to do that. In fact eventually Later on this course we will talk we will be talking about parametrization of surfaces but for now we try to choose the more economical way and a more economical way for a plane is instead of writing a parametric form To write down one equation and put that equation instead of trying to determine the plane by things which belong to it We determine a plane by something which is orthogonal or perpendicular to it by a normal vector Which is our favorite my favorite? Image of this class so far Okay, so very easy to remember So that's a normal vector. So see the point is that there are two vectors There should be two independent vectors determining the plane, but it's also determined by one vector which is perpendicular to it So the parametrization instead of a parametrization We have one equation and the equation has the form a times x minus x 0 plus b times y minus y 0 Plus c times z minus z 0 And I'll draw it here So that's the plane and that's a normal vector. I draw it in a different color It's not a direction vector. It's perpendicular It's perpendicular to the plane. That's why we call it n We call it n normal vector and That's the vector ABC So this are the data ABC which are given The the coordinates of the normal vector components of the normal vector and again, there is a particular point chosen x 0 y 0 and Z 0 as before You get this equation by looking at the by using dot product so to derive this equation you use the odd product but at the end of the day what you get is this is this kind of equation and this lets you write Describe mathematically algebraically describe a given plane once you know point and and the normal vector and There are various ways in practice to find those data from the information which is available For instance a typical problem on homework is like this suppose you are given three points and That there is a unique plane if the points are in generic position There is a unique plane which passes through all of them and the question is to write down the question of the sword How to do that? Well You need one point and one and one normal vector. So you have three points choose one of them and then you need to Find a normal vector and you know you find the normal vector by taking a cross product of two vectors Which belong to the plane which you can easily find by subtracting coordinates of this of these points So you get this n is a cross product of this guy and this guy So that's the way you do it You just obtain the information needed for this formula by using the information which is given Are there any questions about this? Okay, good So what's next? So next we'd like to understand more complicated objects in in the three-dimensional space in R3 and And we start with with with two-dimensional options. So You want to look at more complicated? more complicated surfaces in R3 The question is what's the next example to consider? Okay, so here we have a plane we understand the planes Fairly well. It's a very very simple equation. And so what's so important? What is an important feature of this equation? Well, if you look at this formula, it's actually a good idea to open the brackets Sometimes it's good idea to open the brackets and and really think of this as a function of x y and z on the left hand side So if you open the brackets You open the brackets you get a x plus b y plus c z plus You're going to get a combination of a x 0 and so on Which I will I would like to convert into in one into one symbol Which I'll call D so D is just negative a x 0 negative y negative b y 0 negative c z 0 But it's kind of long so but but the point is that all of these numbers are given These are all of them are given so it is a number unlike x Y and z maybe maybe It's good to emphasize that by using different chalk for this. So you see the very important point is These are variables These are variables and these are numbers. They are given in any given problem. This will be some particular numbers So one two three five whatever So we look at this we look at this formula and we see that on the left-hand side The left-hand side is a function of three variables x y z of the simplest possible kind possible kind Means that each variable enters in degree at most one At most one so it's a polynomial of degree less of degree one It's a polynomial of degree So what I'm trying to say is that? Suppose we were just just for the sake of it We were to try to write down various function in x y and z So what are the simplest things that we could possibly write? Well, first thing is a constant function It's kind of this is sort of a trivial example in some sense a constant function or maybe Any number not necessarily one you could write square root of two or pi or three or ten whatever But one is this is the number it is not it's independent of the variables. So that's the simplest function At the next level, we've got x y and z These are Monomials in x y z of degree one at the next level we can form x squared Y squared and z squared and we can also have a mixed combinations like x y x z and y z so these are monomials of degree two These are monomials of degree two and this are of degree one And if you want this is of degree zero and of course you can continue So next you would write x cube Y cube And then you'll have mixed terms of different of different kind It's actually it's a very interesting question to see how how the number of this independent monomial grows As the degree grows it actually grows very fast It's going to grow very fast So it's actually good a good exercise to find a formula for the number of independent monomials So you see here you have one then you have three then you have six. What's the next number? So anyway, you don't have to do it now, but think about it. It's a good. It's a good question So If we if we are if we want to be methodical One way to approach the question about general general functions in three variables or General surfaces in R3, which are essentially very close Questions which are close to each other then we should start with the simplest ones with surfaces defined by the simplest functions and then progress and include more and more complicated ones, of course We're not going to go all the way, you know, two three four up to ten But at least if we want to do the next possible example, we might as well just go to the next step and next level in this picture So what we've done so far and that's my point is this the first two levels because the most general expression which involves these four guys the constant function the Monomials of degree one is precisely was written on the left-hand side of this equation So using that expression as an equation you get the simplest possible equation that you can write on three variables and sure enough it gives you the simplest possible surface namely playing a plane So now if you would like to continue and go to the next level we should include Parenomials monomials of degree two Okay And this way we get what's called the quadratic surfaces Which is one of the subjects of today's lecture quadratic surfaces, so the idea is To include All monomials of degrees Zero one and two and that should be viewed as a natural generalization of the planes Which include all monomials of degrees zero and one so the question is what kind of surfaces do we get this way? What do they look like so this should give us a good Set of examples which might be convenient in the future when we talk about general surfaces We can test things not we'll be able to test things not only by using planes But also using those quadratic surfaces. That is the general idea now another general idea is When you get a problem in our three Try to do it in our two Maybe even are one in other words if you have a problem in the three-dimensional space try to look at the kind of a baby version of That problem in a smaller dimensional space, which would be in this case a plane The problem this problem is already meaningful or a similar problem is already meaningful in our two so look at let's look at Look first at the analogous problem in our two. We only have two variables X and Y Okay, and so it's easier to analyze What other corresponding curves? I remind you that in our two Because it's two-dimensional if we impose one equation we are going to end up with a one-dimensional object Because two minus one is one right so in our two we can also impose one equation of the form Linear combination of all monomials of degree zero one and two so that would be X squared Let me keep using the red the red color X squared Y squared and then you've got XY and Then you also have a constant So you'll have some a X squared plus B Y squared plus C XY sorry, and of course you also have the ones of degree one so you have plus X so the sum D plus some E Y plus some So it's a little bit easier because there are fewer fewer monomials in two variables You see I only have two of degree one and you have three of degree of degree two as opposed to three and six respectively in dimension three So what does this represent? What does this equation represent? Well, the point is that it looks like there are too many possibilities because there are seven three parameters a b c d e six seven six Six so Too many right, but of course the let's let's let's think about this way we can always If for example, we we can combine some expressions into a square Then we can always say that by changing variables slightly Well, we'll get we will eliminate some some parameters. So what I mean to say is the following say This is something that we looked at before when we talk about circles Suppose you have an equation like this X minus one squared plus Y plus two squared equals one right So if you open the brackets you end up with like x squared minus 2x plus one plus y squared Plus 4y plus 4 plus one So, you know, then you bring you bring the terms and so it's like x squared minus 2x plus Let's write first the second degree Plus y squared minus 2x plus 4y and then you have one plus four minus one So plus four So see this is an expression like this the only thing is so it's almost like a most general expression except there is no Term is x y but you've got x squared y squared x and y You see it looks very it looks quite complicated But the point is that if you complete This if you complete this to square and this to square you actually end up with something much more manageable because We can say let's introduce new coordinates x prime Which is x minus one and y prime which is y plus two Okay, then if we do that in substitute Then we actually then we actually will end up with x prime Squared plus y prime squared equals one and that's a circle, right? That's a circle of radius one But with respect to this new coordinates So what does it mean for the original problem? Well for the original problem It simply means that we have shifted the origin in the in the xy plane. This is the original xy plane and We introduce new coordinates x prime and y prime so for example when x is one That corresponds to x prime equals zero So what we've done is we've introduced new a new coordinate system by just simply shifting The the old axis we've shifted the y axis by one because now This this is x equal one but x equals one corresponds to x prime Equal zero and likewise We have the second line here which corresponds to y equals negative two Which is nothing but y prime Equal zero so we've got a new coordinate system with New coordinates x prime and y prime Right and the point is that the circle this is the equation of the circle in this new coordinate system Where the circle has as the center the origin of this new coordinate system. So it looks like this That's the circle we are talking about but if we can understand it in a new coordinate system That's surely and surely we understand it in the old coordinate system because what it means simply Is that it is a still a circle? But it's a circle centered instead of the origin of the old coordinate system centered at the point at the point Let me write it in white emphasizing that this are the coordinates in the old coordinate system one and two one and negative the center Do you see what I mean? Are there any questions about this? By by choosing a slightly better coordinates you get a much better expression for for your equation Okay, so then the question is really not so much to understand What each of these equations gives rise to but what is the simplest form to which we can bring this equation? By making a similar coordinate change. So what kind of coordinate changes are allowed? First of all shifts are allowed like this x goes to x minus 1 y goes y plus 2 that certainly is it should be allowed Because I mean we don't lose anything clearly we can work with this coordinate system in as much as we can work with this coordinate system The other thing which we should allow is rotations Rotations of the plane which would mean the following you have this is your original coordinate system and say you rotate it by 30 degrees five over six and You end up With this coordinate system So again, this is not such big deal because you know think about it If you look at like this you see this coordinate system, but if you look like this you get this one So it's the same thing. It just depends on your point of view. They are equal It's just it's we shouldn't approach things with prejudice and say that it has to be like this because None of there. There's no reason to say that this coordinate system is better than this one Well, there is an important point that When we rotate when we rotate we preserve the angles and distances so the essential characteristics of geometry are preserved and then in this sense We should not really worry too much if we can get a better shape of the equation by making a rotation So all of this was to say that even though the original equation if you write it in the most general form looks very Complicated you can choose you can always choose a nice coordinate system to bring it to a much simpler form And so what are these simpler forms that we can get? We call it we call it sometimes the canonical form so the canonical form for Some other variables in other words there always exist some variables in which you will get the canonical form now Those variables would would be more appropriate to call X prime and Y prime But I don't want to make a formula look too heavy So I will use again X and Y even though with respect to the original equations This will not be X and Y, but it will be some X prime and Y prime Okay, so what are the what are the? What are the possibilities the first possibility it looks like this X squared over a squared plus Y squared over B squared equals 1 This is something we have encountered before when we talked about curves on the plane This isn't this is what's called an ellipse Because see the point is we surely know very well what the picture looks like if A and B are both equal to 1 It's again the circle of radius 1 which we keep talking about Which we understand very well What we've done now is we've divided the coordinates X and Y by A and B which simply corresponds Geometrically to kind of squishing or expanding depending on whether a is less or greater than 1 Expanding the picture along that axis So the result of this is not a circle, but something which you get sort of a Squish circle Let's let me draw or let me draw a picture for it This this is what it will look like if a is greater than B if a is less than B It will be squished in this way And in this picture, this is a and this is negative a and this is B And this is negative B why because if you substitute X equal 1 you get from this Term you get 1 and then you substitute Y equals 0 and you get the equation So surely this point belongs to it and for the same reason this point belongs to it And similarly if X is 0 and Y is plus or minus B These two points We also get an equality So that's an ellipse now the second the second way the second possibility is The second possibility is a like this X squared over a minus Y squared minus Y squared over B Is equal to 1 and that's that's that's called a hyperbole Square So what does it look like? You see again, I can plot a point where it intersects the X axis I can take a here and I can take negative a and See this is great because if you take a Square over a squared minus zero. This is one So this is like those points, but you cannot do the same as before because if you substitute B Instead of Y you get B squared over B squared, but you have a negative sign. So that's the key difference Here you have plus and actually I would emphasize that here I have plus in both terms and here you have minus And because you have minus you have minus here and so this is not equal to 1 anymore. It's equal to negative 1 right So those two points will not show up And in fact instead what the what the curve will look like it will look like this Now if you think that you're not familiar with this you're mistaken You are familiar and because we have studied hyperbole's and hyperbole's But usually we write hyperbole's by the equation Y equals 1 over X or maybe some coefficient You know C over X you see What we used to call hyperbole before is given by this equation. It's a graph of the function 1 over X right This is why Y equals 1 over X the way I drew it it looks like it's going to intersect The X axis, but it's not right. It's called what it's what's called asymptotic So there are two asymptotic lines to this graph, which are the coordinate axis So this is a very familiar picture. So what's the connection between this picture and this picture? Just to understand this connection Let me let me talk about the special case when a and b are equal to 1 and the general case is very similar When a and b are equal to 1 what this looks like is y squared minus y squared equals 1 Now here's a trick. I Can write this as x minus y times x plus y and now I'm going to perform A kind of transformation that I talked about here Namely, I will choose new coordinates x prime, which is x negative x minus y and y prime which is x plus y if you make this transformation then this equation becomes x prime times y prime is equal to 1 Which is the same as to say that y prime is 1 over x prime Which is our old equation for the hyperbole? So you see at first glance there is no connection whatsoever between this formula and this formula But there is a connection and you do realize this connection by using this transformation Now what does this transformation represent? This transformation actually represents rotation by 45 degrees. I'm simplifying things a little bit because actually what it is it's not just rotation but also It's a composition of the rotation by 45 degrees and also multiplication of everything by by a factor of square root of 2 So there is a square root of 2 here Which is due to the fact that the cosine and sine of pi over 4 is 1 over square root of 2 But it's it's it's beside the point here. Let's not worry about this It's a minor issue at the moment What is important is that there is it if I make a transformation like this which is a lot essentially rotation by 45 degrees I Bring this form to this form Okay, and so that's the power of this kind of transformations Which it allows you to to relate equations which at first glance look look very different But in this particular case we can actually use we can use this transformation to take advantage of our knowledge from before Our knowledge of the hyperbole to understand what the graph what sorry what this curve looks like in this case Because what it looks like is just the old hyperbole rotated by 45 degrees And if you rotate it by 45 degrees, that's exactly what you get rotate this that picture Is a rotation of this picture by 45 degrees clockwise? Yes Why is it clockwise because you have to when you look at this formula you have to explain why this rotation by 45 degrees And not by negative 45 degrees So this is something by the way, which is a subject of another mathematical course called math with 54 And I know that some of you may have taken it or planning to take it So that's the course where this kind of stuff will be discussed in a very systematic way here. We're not going to Really dwell on it too much. I'm just giving you This as an example of the advantages of changes of coordinates and also by way of explaining why this picture Will appear if you want to study that equation This is the explanation because I have reduced the question of drawing this curve to the question of drawing this curve Which we already knew so this are not the only possible cases. This are not the only possible scenarios for For this quadratic curves This is case two So there is another important case Which I'll call case three You see the point is that up to now in the left-hand side I only had monomials of degree two x squared and y squared. I Did not have monomials of degree one and the third case is a case when you do have a monomial of degree one And that's the case when you have Which I'll write like this y Plus in this case. It doesn't matter y plus x squared over over a square Is equal to zero and now that's that's also something which we know very well. That's a parabola Because I actually wrote it in such a way that we can we can quickly Recognize here the graph of a function minus x squared over a square so and So that's this parabola. That's this parabola and also if you want you also you can you can add The other one where you would have y minus x squared over a squared Equal zero so that's the parabola. So this is red and this will be yellow because that's like y equals So two parabolas one going upward the other one going downward In some sense you can argue that these two equations are also equivalent because you can get one equation from the other By flipping the sign of y you put negative y it's the same as putting a minus sign here So it becomes a subtle issue as to which coordinate changes do you allow do they have to preserve orientation or not? So I don't want to get into this So if you want to think think that there are two different cases sub cases here Parable of pointing upward and parable pointing downward So this are the generic This are the generic quadratic surface it quadratic curves. I'm sorry because we are on the plane So these are quadratic curves These are not all of course because if I really insist If I really insist on the most general equation like this I might as well take an equation in which there are no quadratic terms whatsoever And there are only linear terms, but if I do that I go back to the case of degree one Or degree zero and one more precisely and that's the case of a line So I end up with a line. We have already discussed lines. So We're not losing any generality here by assuming that actually there are some non-trivial quadratic terms in the equation Then you get one of those three cases So all of this was by way of illustrating what we are up against now when we would like to understand a similar similar question in in space in R3 instead of R2 See in R2, it's easier to explain there are only there are these three cases But now we need to generalize it to the case of the three-dimensional space, which was our original problem After all right now we are dealing with R3 and we trying to understand curves and surfaces in R3 This was a good digression though. Anyway, because it also gave us some useful information about curves on the plane So in R3 we have to also include Include our third variable z squared which will also give rise to To the two cross terms like this Xz and Yz with some coefficients, so it will be some ABCDEF And then there will be some linear term So there's a whole bunch of additional terms additional terms we have to include However, as I already explained in the case of a plane We want to transform this equation to the simplest possible form on the plane We get these three cases and in space we are going to get the following cases, which I'm going to explain now Okay, so again transform to a nicer expression by using a Different coordinate system that would be kind of a canonical form Canonical form for quadratic surface. So what are these canonical forms? They look very similar. They look very similar to what we had on on the plane the first few the first few cases First few cases will be just like on the plane On the left-hand side We're going to get it have a sum of squares or combination of squares with different with different signs plus or minus so the first one is when all signs are plus and It's very easy to understand what this is just like the way we understood just the way we understood the ellipse Because there's a special case which we understand very well The special case when a is b is and b and c are all equal to one Then we get the equation x squared plus y squared plus this squared equals one And that of course is a sphere of radius one which is centered at the origin. This is something we discussed before So we already know sphere. So when I said we only know planes. I actually wasn't it wasn't quite quite true We know we already talked about spheres So the sphere is a special case of this But not only it's a special case of this But this is a case which will help us to understand the general case the way we were able to derive the ellipse from a from a circle Because what we see is that now by dividing these coordinates by a b and c We have basically just squished our sphere in a certain way in each of the three or expanded if you will Depending on whether a b and c are greater more than one or less than one we shrunk or expanded the sphere In each of the three directions and so what you get it kind is a kind of a it's a kind of a nag I don't know Right. So that that's that's a that's a surface. It's called ellipsoid You take the you take the names for the quadratic curves and you you change the SE at the end by SOID No, not as Anyway, you you you change the hand by oh it That's that's how the terminology will be formed. So you have ellipsoid hyperboloid and Paraboloid. Oh, by the way, I forgot to say the right. This is a parabola and we'll get paraboloids in space Okay, so What what does it look like? It looks like this. It looks like a sphere well To be able to draw this you have to know how to draw a sphere to begin with and then you kind of try to squish to squish everything a little bit I Don't know that's my interpretation This is a kind of contemporary art interpretation of an ellipsoid Think of an egg But the egg is not quite symmetrical usually this has a higher degree of symmetry Okay, I will not do I will not try to make it better you you do it you try you try to get a better picture I think it's pretty I think it's pretty clear. So what's next? Well clearly just by analogy With with with a two-dimensional case. We need to allow some of the signs to be negative Okay, and then there are there are several options You see you've got you've got yourself three signs In front of x squared in front of y squared in front of z squared And so you can play with them any way you like and so let's see what the results are So you've got z squared now over c squared and this is equal to one Now that the point is the problem is in two dimensions. We only had two choices plus or minus you see At first glance it looks there are two more choices because you could have minus plus or minus minus But if you have minus plus It's the same as this one Plus minus if you just relabel the variables x and y You just relabel the variables. So it's not essentially new That's number one number two You cannot have minus minus because minus x squared minus y squared Well, even if you divide by whatever a squared and b squared This is going to be negative always or zero it will never be equal to one and on the right hand side We purposefully put one So we exclude that possibility that all of them are negative So some of them have to be positive and some of them have to be negative And what counts really is not which ones are which but how many pluses and how many science? minuses Here all three are pluses. So the two remain in cases are when When When there are two pluses and one minus so that's sort of like case two Point one And then you have the case two point two Which is where when you have one plus and two minuses So that would be x squared over a squared Plus Minus sorry, so this is plus always So the question is what do you get in those cases and Here to analyze to analyze this what is useful is to to consider what's called sections I will illustrate this notion in the case of an ellipsoid Even though my drawing is not that great, but I will try to explain it So see this is a this is a surface which is like a sphere You know what? Let's just look at the sphere. Maybe it's better so that I can emphasize the most important aspect of a section without Straining our eyes with an ellipsoid So a sphere So this is our coordinate system, okay, and this is a sphere Let's see if I can do it. So this is a circle and this is like this is like the Equator and this is the equator on the other side. It's more or less looks like us like us like a sphere Okay, that's a sphere So now what is what exactly is this equator? What exactly is the equator? I claim that the equator is nothing but what you get by intersecting the sphere with the xy plane Remember, this is the xy plane If you if you cut the sphere with the xy plane what you'll get is precisely this equator So the original equation was x squared plus y squared plus z squared equals one now cut by the plane by the xy plane and The equation of the xy plane is z equals zero, right? It's x y plane So x and y could be arbitrary, but z is equal to zero x y plane, but z is equal to zero So if you cut by z equals zero it simply means that you substitute z equals zero into the equation And so you end up with the equation x squared plus y squared equals one and that's a circle And sure enough we see that that's the equator which we are used to which we knew was there to begin with now the interesting point is You you can cut by other surf other planes as well We can cut by a plane for example z equals one half So that's a plane which is parallel to the xy plane, but it's just it sits a little bit above it at the height of one half When we do that we are going to cut a smaller circle Which is like one of the if you think about this as the earth as the globe as a globe this will be the parallel one of the parallels and So on so you so that you can think of in fact you can think of a sphere of the entire sphere as a collection of those sections As you you can cut by those you can slice it by those planes and each time you slice it you end up with a circle So if we didn't know what the sphere look like We could just imagine the collection of those slices and kind of put them together Into it's like if you don't know what the loaf of bread it looks like but you have a collection of Of the slices you can kind of reconstruct in your mind what the bread must have looked like by kind of imagining those slices On top of each other or side by side so that's the idea of understanding surfaces You have to realize that the surface is really a collection of curves Those curves are obtained by slicing it by parallel planes for example z equals a constant or Instead of z equals a constant you could slice it by x equals a constant or y is equal to constant And that's the way we can analyze this picture and each time we slice it Each time the slice it we are going to end up with a curve and that curve will be one of those familiar curves Which we have already discussed so this is the way to Understand what those surfaces look like you try to to see what what the sections what the slices are and recognize you recognize the slices simply because you have already You have already understood what what the quadratic curves look like So But I don't want to spend the whole hour explaining to you every detail of this So I'll just give you the answer I will just draw you a picture what it looks like and then I will justify it by a similar argument Okay, so the first one the first one that I will draw will be the picture for For the case 2.1 where point one where You have two pluses in one minus Okay, so what does it look like I claim that it looks as follows So this is a parabola which I had before And then I rotate this parabola around Around the z-axis so it's going to look like this Okay, you get the idea and the reason is very simple So of course, I mean the usual corner I don't want to draw the coordinate system on top of it because it's already pretty messy So I don't want to make it more messy So I'll just draw it on the side, but you should think that well the z-axis is here I'll just let me just draw the z-axis. This is the z-axis but So this is the X and Y. It's a usual coordinate This is the z-axis So how do I know that it's like this? Well, the point is if you substitute some value of Z You see it's going to be minus z squared Over c squared equals one, but then I take it to the side. So I get one plus z squared over c squared so I get some positive number and I get the equation x squared Divide by a squared plus y squared divided by b squared is equal to this I now already know that that is an ellipse. So that means that my surface is going to be a collection of ellipses which The smallest which the smallest size will be when z is equal to zero and then it sort of grows in size when z becomes larger Positive number large positive number or large negative number So that's how I know and finally I can also look at the I can slice it by by a zx plane And when I slice it by zx plane that means I set y equals zero and then I recognize hyperbola So in this picture you recognize two old pictures at the same time You recognize two old pictures. One is an ellipse That's an ellipse which you get by setting z equals zero ellipse But you also recognize in this picture hyperbola Which you get by setting by setting x is equal to zero and that's a hyperbola Well, it's not really a circle because at z equals zero I'm going to get x squared over a squared plus y squared over b squared equals one right and we have already decided that this is an ellipse It's almost like a circle and for the purposes of this diagram for this picture in the approximation Which I use it looks like almost like a circle But we have to realize whenever whenever I draw a circle I include all possible ellipses as well because they all they look so much alike Other lips and a hyperbola look very different. So then I would not mistake one for the other I should not mistake one for the other but the ellipse and the circle in first approximation is almost the same So in my picture, I will not distinguish between them So this is a diagram This is a surface which combines an ellipse and a hyperbola and that's why it's called an an an elliptic hyperboloid elliptic hyperboloid That's right So I was going to keep it in suspense because then I was going to make another one and say what are we going to do? Because I have two different names, but now you spoil it. So that's okay. It has one sheet, which it's connected This is just one piece and so then this already suggests that In the second case, we're actually going to get more than one So we're going to get in the second case, we are going to get We're going to get two pieces, two sheets So in the second case, I'll just draw the picture Without explanation, but explanations are obtained the same way So in the second case, you see I'm going to have two I'm going to have two parabolas Sorry, two hyperbolas X, Y and X, Z. I'm going to have hyperbolas and So it's going to look let me see what's the best way to draw it Okay, so it is like taking a hyperbola and rotating it, but in a different way So That's right, that's right. So So This is now actually like I will draw the coordinate system because it's not going to mess up the picture too much This is X, Y, Z So this is a hyper this two things are hyperbolas this this at the two branches of the hyperbola, which is It includes Z and Y. I know I didn't do it right. I Didn't do it right. Yeah. Yeah. Yeah You can tell What I did what I wanted to do You see what you have what in my picture X and Z if you look from the point of view of X and Z then you'll get ellipse ellipses So X and Z should have the same sign and not so it's like this So the hyperbola the high this hyperbola is in the Z Y plane So to get it I have to I have to just retain the Z and Y part of the picture and that's Y squared And I well I got this part right. Okay, at least that's good So this is X equals zero You see so this is this is the case when X is equal to zero I just set it because I get this and that equation is is this red is a red one That's this one. Okay Again, I will not I don't have time to explain the detail of this You'll have to figure this out why it's like this and I will draw you one more picture because we have to also tackle paraboloids and there are there are two types of paraboloids an elliptic paraboloid and And a hyperbolic paraboloid an elliptic paraboloid is kind of easy. It's just like rotating the parabola around the Z axis But the really cool one is is the hyperbolic paraboloid That one is the hardest one to draw. I used to I used to do very good pictures of that one Let's see if I can do it. Let me just let me just draw it and then we'll move on So it's like a saddle you see there's a saddle point You see what I mean, it's not so better. Thank you So Anytime anytime you need a hyperbolic paraboloid just call me Except except on the exams. I will not I will not make you do it. So don't worry All right, but you should be able to visualize it. So that's that's a hyperbolic paraboloid now So you see the the upshot of all this is that we get a Variety of surfaces by simply going from equations involving just degree one monomials two equations Which involve degree two monomials or and lower and so that's a very interesting That's a very interesting example, which kind of indicates the variety the huge variety of surfaces that we have in three-dimensional space Okay, so that's that's what we wanted to see in the case of surfaces Okay, and now we go back to two curves again the whole the whole game that we are playing here in in the three space in the three-dimensional space is It's a kind of interplay between surfaces and curves and You have a question. What's the equation that makes that? Okay, let's see so this is that's actually I'm glad you asked because Because otherwise we kind of missing the punch line. So that's the way this right there is XYZ So it's something which should look like So first of all, it should something which should look like a parabola in this direction if I look on the XY And it should look like Sorry, this should be hyperbole this is a parabola and this is a parabola So first of all, I want to have Z and I want to have In the Y plane I want to have let me try different colors to kind of indicate you So this this green Okay, this green green stuff is nothing but y equals Sorry Z equals y squared Let's forget about this ABC the coefficients. We can we can divide by those coefficients later I'll just I'll just write write write down the rough formula for it So this is equal Y squared. So that should be part of the equation this one Right, this is like the parabola. I still haven't erased it. It's right there What else do we have we have this part of the picture? Which is also parabola Which is also parabola this one is But it lies in So this one was in the Z Y plane and this is going to be in the X Z plane you see in the X Z plane Maybe I didn't draw the Didn't draw this coordinate in my cell I'm actually doing it like this so that's that's the that's the Z equals negative X squared And so what it means is that you should have Z Right, you should have Z you should have negative Y squared and then you have X you should have X squared That's what that's what it should be but Because see this what I did is if I set X equals 0 I Said X equals 0 I get this one So I get this equation Z equals Y squared and if I said a Y equals 0 I get Z equal to minus X squared So that's all good, right and so And so finally If I said if I said Z equal some constant I'm going to get I'm going to get hyperboles. So that's that's roughly the equation I'm not being precise, but this is roughly the equation So it has a Z and it has X squared and Y squared was but it's opposite signs Okay now so so the next topic that we'll discuss is General curves More general curves in space So more general curves in space and again just like just like for just like for lines We parameterize them instead of writing equations because now the curve has dimension dimension of the curve is one Right, so that means we need one parameter But two equations or two equations because the curve lives in a three-dimensional space So each time we write an equation with dimension drops by one It has to drop from three to one therefore we have to write two equations But if you use parameters, we need only one parameter. So it's more economical parameterization instead of writing down equations so we introduce a new a new auxiliary parameter which Usually we call usually we call it T But we don't have to we can call it S or theta whatever whatever letter you like just as long as it's different from the three variables X Y Z and We write X equals F of T Y equals G of T and Z equals H of T and of course we recognize that recognize two special cases here One special case is like this When you don't have the third equation the third formula You only write X equals F of T and Y is equal to G of T. That would be a parametric curve on the plane and We also recognize as a special case the case of a line where F of T is a linear function and so are G and NH So in a sense when we are when we are getting lines in three space, we are also doing the simplest possible parametric functions F, G and H the functions which involve only the constant term and the Degree one term in the coordinate T Now there is there is one there is one terminological issue which I want to emphasize When I say equations I Mean equations just involving X Y and Z Because someone can say well, this is this also looks like an equation So what do I mean when I say you need one parameter or two equations here? I'm actually writing down three equations That's a terminological issue. I don't think of this as equations Because an equation is is something which is a constraint on on the variables which are given to you And this this formula involves not only the coordinates which are given X Y Z, but also some auxiliary Coordinate, okay, so this is not these are not equations Well, we are equations in some sense, but they are not these are not what I mean by equations by equations because they involve involve T an additional variable additional variable Or what we call parameter you see that's why I would like to think of this as parameterization I parameterize X I parameterize Y I parameterize Z I'm not writing down equations on X Y Z an equation on the other hand is like this This is an equation This is an equation It only involves X Y and Z there are no additional variables like T you see that's the difference Is that clear? So don't be confused when I say when I say equation. I don't mean formula like this. I mean formula like this Which involves only the given now someone can say We can actually interpret this and as an equation, but on four variables Because you can think of this now as a system of equations on variables X Y Z and T Which is actually another way to think about this In other words instead of starting in three-dimensional space in trying to parameterize a curve by one parameter You can think that what you are doing is that you are working in a four-dimensional space and you are imposing three equations Because you can look at the four-dimensional space with coordinates X Y Z and T and then you can impose this equations And you see everything is consistent because this way you start with a four-dimensional space and you impose three equations So what's the dimension of the object? It's four minus three because now there are three equations, which is one So it's all consistent you get again one as a dimension of your object But I prefer not to think of this expression as a system of equations in a larger dimensional space Namely in four-dimensional space I would rather prefer to think of this as a way to parameterize my my My curve by this auxiliary parameter actually recording Any questions about this? This is really a terminological issue But I wanted to emphasize it because I keep saying parameterization equations and so on and so I do realize that it could be confusing Okay So what can we do with this more general? What can we do with this more general parametric curves? Well We look at we try to recall what we did with parametric curves on the plane and we try to do the same thing here And one of the things we did was we wrote down equations for the tangent lines To this curves and that's what that's what we'd like to do for curves in a three-dimensional space so let me Let me do it by way of doing an exercise an example Find Parametric equation to the curve Sorry Parametric equation I skipped towards of the tangent line of the tangent line to the curve given in parametric form as follows x is equal to 1 plus 2t y is equal to 1 plus t minus t squared And z is equal to 1 minus t Last squared Minus tq at at the point at the point 111 so how to do this problem? The first thing you need to do is to check whether this point actually belongs belongs to this belongs to this to this curve In this case it clearly does right because this point This point corresponds to the value t equal 1 so it equals 0 If t is equal to 0 we get 111 for all for each x y at z So this of course is a necessary condition because if it's not like it's not Satisfied if there is no parameter t for which you get precisely those coordinates It means that there is a there is a misprint in the problem. So then you cannot really do it. It's not self-consistent So the first step is to find the value of t Step one is to find the value of t Which corresponds to this point and the way you find it is by solving the equation for example here It will be 1 plus 2t is equal to 1 Which gives you t is equal to 0 and then you hope You expect that the problem is correct that if you plug that solution to the to the formulas for y and z you will get the Right values in general it could be that there are there's more than one solution that you get from the first equation So then you have to to find which one will correspond to this point You know one in other words for which value of t where you will actually get that particular point It could happen also that there are two different values of parameter t Which give you the same point that's called self-intersection. We've seen that when we when we were graphing Curves especially by using polar coordinates, so that's also possible but if If that's how it happens then the question would be find parametric equations of the tangent lines Not just tangent line because in this case there will be two different tangent lines This one and this one so The question indicates that actually this point is not a point of self-intersection But rather it's a point like this where you actually have a well-defined unique tangent line Okay, so that's step one to figure this out step two Is to write down the equation of the line and now we have to remember what information we need to write down The parametric representation for a line in three space. We need a point and we need that a direction vector Unfortunately, we fortunately will already have a point one one one. We already have the point So we need the direction vector of for this line and the direction vector is going to be a tangent vector Through this curve and we find it in the same way in which we found it for curves and on the plane by taking derivatives by making derivatives of Our functions at for this value so in general The formula is that you take F prime of t0 G prime of t0 and H prime of t0 Where t0 is your value? Right, so which we simply have to differentiate each of the three functions and substitute Okay, it's five. Let me take one minute because there is one important point and I don't want to to lose it So you see your your curve is already Your curve is already given by parameterization of some variable t And now you're going to write an equation for a tangent line a tangent line is a different curve So you should use a different coordinate to parameterize it use a different coordinate for the tangent line use a different parameter usually We Can utilize the letter s say T s, but you can use whatever you want So for example in this particular case if you take the derivatives the value of the derivatives at equal zero you will get two one and negative one and so the equation of the tangent line is going to be one plus two s one plus s and Z is one minus That these are the equations of tension. Okay, so we'll continue on Thursday