12.6: Cylinders & Quadric Surfaces

okay so we're gonna start with the cylinder cylinder is gonna be a little bit different than what you guys have known before for a cylinder definition of a cylinder a cylinder is a surface that consists of all lines parallel to a given line and pass through a given plane curve okay so I know that that doesn't make a whole lot of sense so I will tell you in simpler terms what that means to us what that means to us is that when we are in 3-space the equation of a cylinder has only two different variables so cylinder us in 3-space is any equation that has two variables rather than a three so for our first example an example of a cylinder would be the equation Z equals x squared so this is 3-space only two equations we did a little bit of sketching these before do you guys remember how I suggested that you sketch them I suggested doing two space first and then go into three space so I would suggest that you still do that here here we are plotting X versus Z it's a parabola okay so what this shows us then because why is that involved when we move to three space we're gonna be taking this parabola and moving it down the y-axis does that make sense so it's gonna end up looking something like this I will tell you I'm not an artist so I will do the best I can to draw these if you can do a better job great okay I'm gonna start by sketching this parabola we're just y equals zero so that's this one and then this needs to continue along the y axis do you get the gist of what I'm going for in my picture yes I mean I showed you in real life so anyway this is called a parabolic cylinder so that's why I said it's not exactly the cylinder that you guys know this doesn't look like a typical cylinder this is still a cylinder though because it only has the two variables so we'll do one more example before we go on to quadric surfaces next example is y squared plus Z squared this one so again sketch it in two space first Y versus Z doesn't matter where you put them but it is a circle okay so then what's gonna happen here because X is not involved X or because X is not in the equation X is gonna be the axis that we move this circle along so we're gonna take the circle move it along the x axis and what are we gonna get a cylinder yeah the cylinder that you guys typically know this circle so that is our cylinder should this should not be new because we did this I think the very first day or the second day you ready to get into more interesting graphs great next thing we're talk about is quadric surfaces ok quadric surfaces are graphs of second degree equations that have three variables so this is the case where it's 2nd degree it's in three space we have all three variables so here is the most general equation ax squared ad be Y squared ad C Z squared ad D XY + ey z ad f x z GX @ h y @ z ad j equals 0 now this is not helpful in any way shape or form so don't worry about it I'm not gonna ask you to memorize this and I'm not gonna give you any equation that looks like that we're gonna look at I think six general shapes okay this is where our drawing skills really need to be good okay so first example or example number three but it's our first quadric surface it's gonna be the surface z equals x squared plus y squared okay so what I'm gonna do with each of these equations is I'm gonna show you different ways to analyze the equation so we know some things about the graph I think that's a better idea than just memorizing everything first thing I want you to notice here is that Z is always gonna be positive x squared Y squared are positive so Z is always gonna be 0 or bigger so that's helpful to us we know that our graph is only gonna be the top now what you've done in the past when you didn't know how to graph an equation as you just plugged in numbers and then you plotted points now you can see here that's not gonna be very helpful because there's no way you're gonna get the whole graph by plugging in it off points so we will not be doing that instead what we're gonna be finding is we're gonna be finding what are called traces what a trace means is you pick values for one of the variables so I'm going to start with Z if I choose Z to be 0 that's going to give me 0 equals x squared plus y squared what is that if we were to graph this equation what would it be it's a point yeah it's just the origin so that gives us this point right here then it might be helpful to choose Z to be 1 we get 1 equals x squared plus y squared what is that circle okay so then at Z equals 1 you're gonna draw a circle with a radius of 1 you could choose Z to be 2 or 3 but Z equals 4 might be a little bit better so we get 4 equals x squared plus y squared again a circle this time of radius 2 these are cool traces okay so then we should be able to see that this is gonna keep happening anytime we pick bigger values of Z we're gonna get bigger circles that just keep going up the z axis so this is what our surface ends up looking like can you picture that it's like a parabola that's being revolved around the z axis do you get the general idea of these okay so here's what we're gonna do then I'm going to take you through all six shapes and then we'll look at a few examples does that work okay so like I said there are six general shapes I'm gonna give you the equation and the shape we're gonna talk about the intercepts and we're going to talk about what those traces are gonna look like first general shape x squared over a squared plus y squared over B squared plus Z squared over C squared equals 1 ok so it's a good general idea with any of these shapes that you find the intercepts so our Z intercept is gonna be when we make X and y 0 if we make X and y 0 the Z intercept will be plus or minus C same idea if I make X 0 and I make Z 0 my y intercept will be plus or minus B then same idea for a plus or X rather plus or minus a comma 0 comma 0 so this has six different intercepts okay now when we did the traces up here I chose Z really I could have made X a certain number I could have made Y a certain number so there are three different chases if you make X some number if you make Y some number if you make Z some number so if we make X some number this is a positive number we would probably move it over and then you would have Y squared over B squared plus Z squared over C squared equals some number what kind of shape would that be an ellipse yes do you guys see that the same thing would happen with y&z okay so all three of these traces are all going to be ellipses okay so then we look at graphing this okay so if we graph our intercepts we have that plus or minus a so let's choose that to be somewhere so maybe here is my a so we have a point there or we have points there we also have our B's that plus or minus B we have those intercepts I'm just picking values of a and B so keep in mind that we don't know what they are maybe C is here that's plus or minus C okay we have a whole bunch of ellipses though so these points right here form an ellipse any pair of intercepts just like the x and z intercepts those will form an ellipse the X and y intercepts will form an ellipse the y and z intercepts will form an ellipse so you need to keep finding all of those ellipses so this is what our figure ends up looking like once you connect all of the ellipses can you guys picture this it's kind of like a 3d oval this figure is called an ellipsoid okay that's first one we have five four so that's first one second one Z squared over C squared is equal to x squared over a squared plus y squared over B squared so again we're gonna talk about those intercepts and the traces okay intercepts there's only one and that would be the origin 0 0 0 if I make X and y 0 the only way that can be the case is if Z is 0 and keep going for our traces X can be some number Y can be some number Z can be some number we start with Z that's going to be the easiest one if I make Z some number what is my equation that I have left what shape can I make z some number so I have x squared plus a squared plus y squared or B squared equals some number that's an ellipse okay these two are gonna be the same if I make X some number so I have y squared over B squared plus some number equals Z squared over C squared hyperbola okay here's what this one is gonna look like whatever is the variable that's by itself is the axis that we will be using this figure is gonna form two different cones one cone that opens up one cone that opens down so this one shockingly it's gonna be called a comb it might be called a an elliptic cone sometimes so it depends on what these are if they're circles or if they're ellipses so if you see a lipstick cone that means that all of these are ellipses next one exactly the same except instead of this being squared it's just to the first power so Z over C equals x squared over a squared plus y squared over B squared so same intercept 0 0 0 looking at the traces if Z is some number we still have x squared over a squared plus y squared over B squared so this will still be an ellipse X&Y being equal to some number that changes if I make X some number I end up with a parabola so those two both give us parabolas so this is very similar to that example that we did the example number three example number three we did Z equals x squared plus y squared so it's going to give us a very similar shape this again is the axis that we move along so as we already saw it looks something like this this is called an elliptic paraboloid so the paraboloid should make sense you see a parabola shape it's called elliptic in the general form these traces are going to be ellipses okay halfway through the general shapes ready for the next ones great looking for you on your homework since it's on WebAssign I can't actually ask you to sketch anything so your homework is mostly gonna be this is the equation pick the right shape your packet I'm gonna ask you to sketch them each though so you will have them here's your next one we have x squared over a squared plus y squared over B squared minus Z squared over C squared equals one intercepts you can find I'm not gonna find these right now I'm gonna just tell you what the traces are if Z equals some number it's gonna give us an ellipse when x and y equals some number it's gonna give us hyperbolas okay here's what this one is gonna look like first of all this one- again that's our axis you're gonna have a small circle at the origin as you get further away from the origin your circle is gonna get bigger looks something like this bless you this one is called a hyper boid of one sheet that one sheet is Nexus necessary cuz the next example we are gonna do is a hyperboloid of two sheets here's how you remember the one sheet one sheet has one negative so when we get to two sheets which we're gonna do now there's gonna be two negatives we have negative x squared over a squared minus y squared over B squared plus Z squared over C squared equals 1 this is a hyperboloid of two sheets two sheets because that's two negatives the way to remember which one is the axis it's always the one that's different so one that the one that's different here is Z it's the only positive one this will be our axis then traces are the same as a hyperboloid of one sheet intercepts you can find yourself we're gonna have an intercept when Z is positive C and when Z is negative C it's gonna look something like this this is a hyperboloid of two sheets ready for the last one okay I left the best one for last general equation is e over C equals x squared over a squared subtract y squared over B squared intercepts you can find on your own when x and y are some number you should be able to see you're gonna get a parabola when Z equals some number what are you gonna get x squared over a squared minus y squared over B squared equals some number what is that hyperbola okay this one is gonna be the hardest one to draw I'm gonna tell you right off of that I'm not an artist okay stop making fun of my drawing it's a saddle like a horse I feel like mine is actually pretty good okay oh so much better okay here's my expectation for you when you are drawing this I expect you to write your intercepts write the traces and draw a general section okay I don't expect you to have necessarily the right intercepts or the right size as long as you get across the point that you're drawing a settle okay we're ready to do some examples I'm taking that away I'm sorry you gotta get it perfect example three where we found probably didn't find the intercepts but we found the traces I'm gonna take you through how I approach these problems yes oh I didn't tell you it's called a hyperbolic paraboloid isn't that a fun one what's your favorite shape the hyperboloid of one sheet [Music] okay so now we're going to talk to talk about like practically how do you draw these how are you going to approach each problem first example y equals x squared plus Z squared over four okay what you could do is you could just memorize all of those shapes in all of their equations personally I think it's that's a waste of your time so I wouldn't suggest you do that but if you want to go for it this is the one that's different that's gonna tell you your axis so our shape will be forming along the y axis first thing that you're gonna find is any intercepts the case will really make X 0 and Z 0 Y has to be 0 okay so we're gonna make XY 0 Z has to be 0 etc so your only intercept is at the origin next thing that you're gonna do is you're gonna consider some of the traces so why I think is the best one to start with if we choose y equals 0 we get 0 equals x squared plus Z squared over 4 which is just the origin so that really wasn't that helpful if we take Y to be 1 what do we get an ellipse okay do we see that as we keep changing the values of Y we're gonna keep getting ellipses okay so that tells us as we go along the y-axis we're gonna get some ellipses so let's start with that okay we have that intercept of zero zero zero when y is one we have an ellipse the ellipse tells us in the X direction we're going to go one and the z direction we're going to go to something like that when y is negative one it's the same thing since you're squaring it it should be the same mine art but it should be let's now consider some X values if we take X to be 0 we're going to end up with y squared equals Z squared over 4 if we take the square root we get plus or minus y equals plus or minus Z over 2 what is that yeah it's two lines that cross it in X so you're gonna consider positive y equals Z over two that's a line negative y equals Z over two that's aligned with the opposite slope opposite sign so it's going to be lines that cross like this as you change the value of X you're still gonna get those lines just moved a little bit so you're getting something like this sorry I know it's not great but hopefully you get the point so you're gonna keep having these ellipses do you see what's happening two cones that are opening up along the y axis so the one we did in the notes was like this up and down because it is along the z axis now we're along the y axis though so again this is called a cone each side those are ellipses so it's often referred to as an elliptic cone okay do you get the general idea of how to approach these yes okay we have two other examples that I want us to do you are also gonna have to identify these quadric surfaces when they are not in the forms that I gave you so for example this next one 4x squared add 4 y squared add Z squared plus 8y minus 4z equals negative 4 so I did not give that to you in any form that you've seen so what is your suggestion to get it in a form that's more useful to us you're gonna have to complete the square ok so 4x squared we're not gonna have to do anything there do you guys remember with completing the square you want a positive 1 y squared okay so for that y squared I'm gonna factor out a 4 I'm have to factor out a 4 from there I'm gonna leave myself space to complete the square completing the square here I take two and I divide it by 2 I get 1 1 squared is 1 now remember though that I factored out a 4 so to this side I'm really adding 4 so I'm going to add 4 to the other side over here I'm gonna have to add 4 at for done so we end up with 4x squared add 4 here that's y plus 1 squared and then Z minus 2 squared equals 4 this is almost a form that we have seen form that we've seen that needs to be a 1 so we're gonna divide by 4 so that we have it set equal to 1 we get x squared plus y plus 1 squared plus Z minus 2 squared over 4 equals 1 okay this I think is the easiest shape to identify can you guys tell before we even find anything what that's gonna be it's an ellipsoid when they're all squared all on the same side all positive it's gonna be an ellipsoid so if we find some intercepts if I make X and y 0 Z has to be 2 if I make X 0 and Z 0 Y has to be that makes Z 0 that ends up being 1 that'll be 0 so Y has to be negative 1 then if I make Y & Z 0 if I make Y & Z 0 here I'm gonna get 1 here I'm gonna get 1 so that does not give us any intercepts so we have just those two intercepts ok all the traces are ellipses okay the kind of vertex of the ellipse or the center of the ellipse is the point 0 negative 1 never mind scratch that okay let's start sketching that's zero negative one zero negative one goes along the y-axis so that's right there zero zero two is right there okay this point 0 negative 1 2 what I was trying to say about that is that's gonna be the center of the ellipse it's not actually part of it but it's where the ellipse is centered so that right there is the center that tells us our next point is up here over here so this is the gist of how our ellipse is going to look okay are you ready for one more any questions okay these are important they might not seem that important but not next chapter really not the following chapter but the last two chapters of calc three are gonna involve all of these different shapes we're gonna be setting up integrals with them to find volume to find surface area things like that so it is important that you're able to recognize these and tell what they're gonna look like last one Z equals y squared over four minus x squared over nine sorry okay in terms of intercepts the only intercept is zero zero zero in terms of our traces if we make X some number we're gonna have a parabola so all of the extra cesare parabolas if we make Y some number same thing we're gonna get parabolas if we make Z some number we're gonna get hyperbolas what figure is that where we're gonna get some parabolas and some parabolas and some hyperbolas our favorite one the hyperbolic parabola yes it is okay I'm gonna do a better job this time I think my picture on here is pretty good so I'm gonna try to get the same picture okay so you're always going to start with this parabola in the middle it's very stressful does that look better than my other one I'm pretty proud of that one that's pretty good for a hyperbolic paraboloid this is what I expect I want to see your intercepts I want to see the traces and then just sketch a hyperbolic paraboloid yes not for these no for any of the other five yes there should be some points that will show the magnitude and the position hyperbolic paraboloid though I don't care about any exact points okay questions

Channel: Alexandra Niedden

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Id: HT32icHGl6k

Length: 37min 8sec (2228 seconds)

Published: Wed Aug 28 2019