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Introduction to Parametrizing a Surface with Two Parameters. Created by Sal Khan.
Video transcript
All the parameterizations we've done so far have been parameterizing a curve using one parameter. What we're going to start doing 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. And we know what a doughnut looks like. Let me draw it in a suitable, well, I don't have any suitable doughnut colors, so I'll just use green. A doughnut looks something like this. It has a hole in the center, And maybe the other side of the doughnut looks something like that, and we could shade it in like that. That is what a doughnut looks like. So how do we construct that using two parameters? So what we want to do, is you can just visualize it, a doughnut, if you were to draw some axes here. So that's our doughut. Let me draw some axes. So let's say I have the z-axis that goes straight up and down. So we have drawn it here, the doughnut's 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 doughnut. 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 coming out of here will be my x-axis. That right there is my x-axis, and then 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 doughnut, I'll draw a little bit neater, but the cross section of this doughnut in the x-z axis, is going to look something like this. If I were to slice it in the x-z axis, it would look something like that. That would be the slice. It would trace out, and we're thinking 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 like that, right there. And if you go out here, you're going to get 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, it'll give us some good intuition for the best way to parameterize this thing. So let's do it that way. 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 y, 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 you cross the z-y axis, the center sits on the y-axis. I didn't draw it that neatly here, but I think you can visualize. So it sits right there on the y-axis. And let's say that it is a distance b away 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. It's always, if you imagine the top of the doughnut, let me draw the top of the doughnut. If you're looking down on a doughnut, let me draw a doughnut right here, 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 doughnut. It will look just like this. And if you imagine the cross section, this circle right here, the top part of the circle if you're looking down, would look just like that. And this distance b is a 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. 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. >From the center of our torus to the center of our circle that defines the torus, 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 a. If I were 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 x-z plane. So one parameter is going to be the angle between our radius and the x-z 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 2 pi, as s goes between 0 in 2 pi, when the 0 is just going to be at this point right here, and then as it goes to 2 pi, you're going to 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. What I just drew is that circle right there. What we want to do is spin the entire circle around. So 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 view. As 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 z-y plane. We are b away from the z-axis, so that distance is b. In this diagram, the z-axis is just popping out at us. It's popping out of the page. We're looking straight down. It's just like the same view as right there. And what I just drew, when s is equal to 0 radians, we're going to be out here, exactly one radius further along the y-axis. And 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 doughnut, we're going to have to rotate this whole setup around the z-axis. Remember, the z-axis is popping out. It's looking straight up at us. 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 this is when we've rotated 0 radians. At some point, 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 our doughnut, right there. At that point, we would have rotated it, let's say p 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 2 pi. And I want to make this very clear. Let's actually draw the domain that we're mapping from to our surface, so that we understand this fully. So let me draw some, and then we'll talk about how we can actually parameterize that into a position vector-valued function. So right here, let's call that the t-axis. That's, remember, how much we're rotated around the z-axis right there. And 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 going to vary between 0 and 2 pi. So this 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 2, this would be 3 pi over 4. You do the same thing on the p-axis. It's going to go up to 2 pi. Let's do that. So we're going to go up to 2 pi. I really want you to visualize this, because then the parameterization, I think, will be 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 zero and 2 pi. And let me do that in magenta, right here. So we're holding s constant, and we're just varying the parameter 2 pi. So this, if you think about it, should just form a curve in three dimensions, 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 my axes. So that is my x-axis, that is my y-axis, and then this is my, I'm getting messier and messier. That is my z-axis, right-- actually, let me draw it a little bit bigger than that. I think it will 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 around this circle at all. That means we're out here. We're going to be b away, and then a away again. Right? We haven't rotated around this at all. We're setting s as equal to zero. So essentially, we're going to be b away, so this is going to be a distance of b away, and then we're going to be another a away. The b is the center of the circle, and 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. These were top views over here. So this line right here, in our s-t domain, we can say, when we map it, or parameterize it, it'll correspond to the curve that's essentially the outer edge of are doughnut. If this is the top view of the doughnut, it will be the outer edge of the doughnut, 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 t-s plane, I guess we could say, this magenta line, we hold s at 0 radians and we increase t, this is t is zero, 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 increase t and hold s constant at 0. Now let's do another point. Let's say when 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 pi, and then rotate it around to form our doughnut. So we're going to form the inside of our doughnut. So when s is at pie, 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 along, holding s at pi, and we increase t, we're going to trace out the inside of our doughnut, that will look something like that. That was my best shot at drawing it. And then we can 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 exactly 90 degrees, right? Pi over 2 is 90 degrees at this point. And then if we vary t, we're essentially tracing out the top of the doughnut, right? So let me make sure I draw it . So the cross section, the top of the doughnut, we're going to start off right over here. So when s is pi over 2, and you vary it right, and then you very t, I'm having trouble drawing straight lines. And then you vary t, it's going to look like this. That's the top of that circle right there. The top of this circle is going to be right there. The top of this circle is going to be right over there. Top of the circle is going to be right over there. So then I just connect the dots. It's going to look something like that. That is the top of our doughnut. If I was doing this top view, it would be the top of the doughnut, just like that. And if I wanted to do the bottom of the doughnut, just to make the picture clear, if I were to make the bottom of the doughnut, the bottom of the doughnut would be-- see, if I take s as 3 pi over 4 and I vary t, that's the bottoms of our doughnuts. So let me draw the circle, it's right there, the circle is right there, you wouldn't be able to see the whole thing if this wasn't transparent. So you'd be tracing out the bottom of the doughnut, just like that. I know this graph is becoming a little confusing, but hopefully you get the idea. When s is 2 pi again, you're going to be back to the outside of the doughnut again. That's also going to be in purple. So that's what happens when we hold the s constant at certain values and vary the t. Now let's do the opposite. What happens if we hold t at 0, and we very the s? So t is 0, that means we haven't rotated at all yet. So we're in the z-y plane. So t is 0, and s will start at 0, and it'll go to 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 will go to 3 pi over 4, 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 pick when t is pi -- let me use 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 now we're over here. And now when we vary s, s will start off over here, and it'll go all the way around like that. So this line corresponds to that circle. We could keep doing it like this. When t is equal to pi, that means we've got all the way around the circle like that, and now 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-z plane. And then when you vary s, s will 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 now of the parameterization. I haven't done any math yet. I haven't actually showed you how to mathematically represent it as a vector value 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 s-t plane correspond to onto this surface, in I guess you could say, in R3, this little square right here, let's see what's bounded by. This little square, I want to make sure I picked a square that I can draw neatly. So this square right here, that is between, when you look at t, it's between 0 and pi over 2. And s is between 0 and pi over 2. So this right here is this part of our torus. If you're looking at it 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 doughnut. 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 going to actually talk about, how do we actually parameterize using these two parameters? Remember, s takes 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 doughnut. 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 three-dimensional position vector-valued function that would define this surface? We're going to do that in the next video.