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Current time:0:00Total duration:19:02

Video transcript

all of the parameterizations we've done so far have been parameterizing a curve using one parameter we're going to start doing in this video is parameterizing a surface in three dimensions using two parameters and we'll start with an example of a torus a torus or more commonly known as a doughnut shape a doughnut shape and we know what a doughnut looks like let me draw it in a suitable I don't have any suitable donut color so I'll just use green so donut look something like this so it has a hole in the center as a hole in the center and maybe the other side of the donut look something like that we could shade it in like that and it may be shade it in like that that is what a donut looks like so how do we construct that using two parameters so what we want to do is you can just visualize that a donut if we were to draw some axes here so that's our donut let me draw some axes so let's say I have the z-axis that goes straight up and down so the way I've drawn it here the donuts a little at a tilt so the z-axis I'll tilt it a little bit so our z-axis goes straight through the center of the donut so that right there this is going to be an exercise in drawing more than anything else so that is my z-axis and then you can imagine the z-axis goes from there and then this on the coming out of here will be my x axis that right there is my x axis and then maybe my Y axis maybe my Y axis comes out like that and the whole reason why I drew it this way is that if you imagine the cross section of this don't I'll draw it a little bit neater but the cross section of this zona and the X z-axis is going to look something like this if I were to just slice it in the XZ axis it would look something like that I'm slicing this is the X z-axis right that and that that would be the slice it would trace out and we're talking about not a full doughnut just the surface of a doughnut so it would trace out a circle like this if you were to cut the doughnut in the positive Z Y axis it's going to trace out a circle that looks something traces out a circle that looks some like that right there and if you go out here you're going to a circle it's just a bunch of circles so if you think about it it's a bunch of circles rotated around the z-axis if you think of it that way we it'll give us some good intuition for the best way to parameterize this thing so let's do it that way so let's start off with just the Z Y axis I'll draw it a little bit neater than I've done here so that is the z axis and that is and that is why just like that and let's say that the center of these circles let's say it lies on you know it can lie when it when you cross the Z Y axis it could the center sits on the y axis I didn't draw it that neatly here but I think you can visualize so it's it's right there on the y axis and let's say that it is a distance B away from the B from the center of the torus or from the z axis it's a distance of B it's always going to be a distance of B all right it's always if you imagine the top of the doughnut let me draw the top of the donut if you're looking down on a doughnut so let me draw a donut right here so if you're looking down on a doughnut it just looks something like that the z axis is just going to be popping straight out the x axis would come down like this and then the y axis would go to the right like that so you can imagine I'm just flying above this I'm sitting on the z axis looking down at the donut it'll look just like this and if you imagine the cross-section if you were this circle right here will look just like that this circle right here that part of the top part of the circle if you're looking down would look just like that and this distance this distance B is the distance from the z-axis to the center of each of these circles so this distance let me draw it in the same color from the center to the center of these circles that is going to be B and it's just going to keep going to the center of the circles B that's going to be B that's going to be B that's going to be B all of from the center of our torus to the center of our circle the define Storrs it's a distance of B so this distance right here that distance right there is B and from B we can imagine we have a radius a radius of length a so these circles have radius of length a so this distance right here is a this distance right here is a this distance right there is a that distance right there is if I were to look at these circles these circles have radius a and what we're going to do is have two parameters one is the angle that this radius makes with the X Z plane so you can imagine the x axis coming out let me do that in the same color you can imagine the x axis coming out here so this is the XZ plane so one parameter is going to be the angle with our rate between our radius and the XZ plane we're going to call that angle or that parameter we're going to call that s and so as s goes between 0 and 2pi as s goes between 0 and 2pi once 0 is just going to be you're going to be at this point right here and then as it goes to 2pi you're going to trace out trace out a circle that looks just like that now we only have one parameter what we want to do is then spin this circle around that's what I just drew is just that circle right there what we want to do is spin the entire circle around so let's define another parameter let's define another parameter we'll call this one T and I'll take the top view again this one's getting a little bit messy let me draw another top views you can see this is all about visualization so let's say this is my x-axis that is my y-axis and we said we started here on the what on the zy plane we are be away from the z axis so that distance is B in this diagram the z axis is just popping out at us is popping out of the page we're looking straight down it's just like the same views right there and what I just drew when s is equal to zero radians we're going to be out here we're going to be exactly one radius further along the y axis then we're going to rotate as we rotate around we're going to rotate and then come all the way over here that's when we're right over there and then come back down so if you looked on the top of the circle it's going to look like that now to make the donut we're going to have to rotate this holes set up around the z-axis from the C axis is popping out it's great looking straight up at us it's it's coming out of your video screen now to rotate it we're going to rotate this circle around the z axis and to do that we'll define a parameter that tells us how much we have rotated it so when we're over so this is when we've rotated zero radians at some point we're going to be over here we're going to be over here and we would have rotated it this is B as well and our circle is going to be looking like this this is maybe this point on the circle on our donut right there at that point we would have rotated it let's say T radians so this parameter of how much have we rotated around the z-axis how much have we gone around that way we're going to call that T and T is also going to vary between 0 and 2pi and I want to make this very clear let's let's actually let's actually draw kind of the domain that we're mapping from to our surface so that we understand this I guess fully so let me draw some and then we'll talk about how we can actually parameterize that into a to a position vector-valued function so right here let's call that the T axis let's remember how how much were rotated around the z-axis right there and let's call this let's call this down here our s axis and I think this will help things out a good bit so when s is equal to 0 and we vary just T so they're both is going to vary between 0 and 2pi so this is right here is 0 this right here is 2 pi let me do some things in between this is pi this would be PI over 2 obviously PI over - this would be 3 PI over 4 you do the same thing on the T axis it's going to go up to 2 pi let's do that so we're going to go up to 2 pi 2 pi I really want you to visualize this because it'll then the parameterization I think will be straight fairly straightforward so that's 2 pi this is PI this is PI over 2 and then this is 3 PI over 4 so let's think about what it looks like if you just hold s constant at 0 and we just vary T between 0 and 2pi and let me do that in magenta right here so we're holding s constant and we're just varying the parameter two pi so this if you think about it should just form a curve in three dimension it's not a surface because we're only varying one parameter right here so let's think about what this is remember s is let me draw it let me draw my axes so that is my x-axis that is my y-axis and then this is my getting Messier and Messier that is my z-axis right actually let me draw it a little bit bigger than that I think it'll help all of our visualizations all right so this is my x axis that is my y axis and then my z axis goes up like that z axis now remember when s is equal to 0 that means we haven't rotated we haven't rotated around this circle at all that means we're out here we're going to be be away and then a away again right we haven't rotated around this at all we're setting s is equal to 0 so essentially we're going to be be away so this is going to be a distance of be away and then we're going to be another a away the radius the B is the center of the circle then we're going to be another a away we're going to be right over there so this is a plus B away and then we're going to vary T remember T was how much we've gone around the z axis this was these were top views over here so this line right here in our st domaine we can say when we map it or you parameterize it it'll correspond to the curve that's essentially the outer edge of our donut if this is the top view of the donut it will be the outer edge of our donut just like that so let me draw the outer edge and to do that a little bit better let me draw the axes in both the positive and the negative domain it might make my graph a little bit easier to visualize things positive and negative domain this is negative Z right there so this line in our TS plane I guess we could say this magenta line where we hold s at 0 radians and we increase T this is T is 0 this is T is equal to 2 pi that's T is equal to pi this is T is equal to 3 PI over 2 all the way back to T is equal to 2 pi this line corresponds to that line as we rotate as we wrote as we increase T and hold s at constant at 0 now let's do another point let's say when s is s is at pi right remember when s is at pi we've gone exactly pi is 180 degrees when s is at pi we've gone exactly 180 degrees around the circle around each of these circles so we're right over there and now let's hold it constant at pine then rotate it around to form our Donuts we're going to form the inside of our donut so when s is at pi and we're going to take T from 0 so when s is PI and T is 0 we're going to be this was the center of our circle we're going to be a below that we're going to be right over there and then as we vary as we increase T so as we move up as we move up along holding s at pi and we increase T we're going to trace out the inside of our donut we're going to trace out the inside of our donut that will look something like that that was my best shot at drawing it and then we could do that multiple times when s is PI over 2 I want to do multiple different colors when s is PI over 2 we've rotated up here we've rotated exactly 90 degrees right PI over 2 is 9 degrees we're at this point and and if we vary T we're essentially tracing out the top of the donut right so let me make sure I draw it so the cross-section the top of the donut we're going to start off right over here so when s is PI over 2 and you vary right and then you vary T I'm having trouble drawing straight lines and then you vary T it's going to look like this is that that's the top of that circle right there the top of this circle it's going to be right there top of this circle is going to be right over there top of this circle is going to be right over there so then I just connect the dots it's going to look something something like that that is the top of our donut if I was doing this top view would be the top of the donut just like that and if I want to do the bottom of the donut just to make the picture clear if I make the bottom of the donut the bottom of the donut would be see if I kick s is 3 PI over 4 and I vary T that's the bottoms of our doughnuts so let me draw the circle so it's right there the circles right there you would need to be able to see the whole thing if this wasn't transparent so take you'd be tracing out the bottom of the donut just like that I know that this graph is becoming a little confusing but hopefully you get the idea when s is 2pi again you're going to be back to the outside of the donut again that's also going to be in purple so that's what happens when we hold the S consonant certain values and vary the T now let's do the opposite what happens if we hold T at 0 and we vary the s so if we hold T at 0 and we vary the S so T is 0 that means we haven't rotated it all yet so we're in the zy plane so T is 0 and s will start at 0 s will start at 0 and it'll go to 2pi it'll go to 2pi this is this point let's start at PI over 2 that's that point over there then it'll go to pi this point is the same thing as that point then it'll go to 3 PI over 4 go to 3 PI 4 and then it'll come back all the way to 2 pi so this line corresponds to this circle right there we could keep doing these if we if if we pick when s is PI over 2 I'm sorry when t is PI over 2 me to a different color that's not different enough when T is PI over 2 just like that we would have rotated around the z-axis 90 degrees so we're now we're over here and now when we vary s s we'll start off over here and it'll go all the way around like that so this line corresponds to that circle I could keep doing it like this when T is equal to PI that means we've gone all the way around the circle like that and then when we vary s from 0 to PI over 2 we're going to start all the way over here and then we're going to vary all the way we're going to go down and hit all those contours that we talked about before and I'll do one more just to kind of make this the scaffold clear this dark purple hopefully you can see it when T is 3 PI over 4 we've rotated all the way so we're in the X we're on the XZ plane we're on the XZ plane and then when you very s s we'll start off over here and as you increase s you're going to go around the circle around the circle just like that and of course when you get all the way back full circle T over PI over 2 that's the same thing you're back over here again so this is going to be we can even shade it the same color and hopefully you're getting a sense of the parameterization I haven't done any math yet I haven't actually showed you how to mathematically represent it as a vector valued function but hopefully you're getting a sense of what it means to parameterize by two parameters and just to get an idea of what these areas on our on our s T on our X s T plane correspond to onto this surface onto this surface in our in in I guess you could say an r3 this little square right here let's see what it's bounded by it's this little square I want to make sure I pick a square that I can draw neatly so this square right here that it is between on when you look at it's between this T is equal to zero and PI over two so between zero is the T between 0 and PI over 2 and s is between 0 and PI over 2 so this right here is this this part of our torus this part of the torus if you're looking at it from an outer edge from our sorry from the top it would look like that right there you can imagine we've transformed this square I haven't even shown you how to do it mathematically yet but we've transformed this square to this part of the donut now I think we've done about as much as I can do on the visualization side I'll stop this video here in the next video we're gonna actually talk about how do we actually parameterize using these two parameters remember s takes us around each of these circles and then T takes us around the z-axis and if you take all of the combinations of s and T you're going to have every point along the surface of this torus or this donut how do you actually go from an S and a T that goes from 0 to 2 pi in both cases and turn it into a position vector da three-dimensional position vector valued function that would define this surface we're gonna do that in the next video