1 00:00:00,000 --> 00:00:00,560 2 00:00:00,560 --> 00:00:03,240 All the parameterizations we've done so far have been 3 00:00:03,240 --> 00:00:05,855 parameterizing a curve using one parameter. 4 00:00:05,855 --> 00:00:09,419 What we're going to start doing this video is parameterizing a 5 00:00:09,419 --> 00:00:12,990 surface in three dimensions, using two parameters. 6 00:00:12,990 --> 00:00:15,870 And we'll start with an example of a torus. 7 00:00:15,869 --> 00:00:22,320 A torus, or more commonly known, as a doughnut shape. 8 00:00:22,320 --> 00:00:25,300 And we know what a doughnut looks like. 9 00:00:25,300 --> 00:00:27,990 Let me draw it in a suitable, well, I don't have any 10 00:00:27,989 --> 00:00:30,929 suitable doughnut colors, so I'll just use green. 11 00:00:30,929 --> 00:00:33,740 A doughnut looks something like this. 12 00:00:33,740 --> 00:00:38,460 It has a hole in the center, And maybe the other side of 13 00:00:38,460 --> 00:00:40,750 the doughnut looks something like that, and we could 14 00:00:40,750 --> 00:00:43,840 shade it in like that. 15 00:00:43,840 --> 00:00:46,640 That is what a doughnut looks like. 16 00:00:46,640 --> 00:00:51,009 So how do we construct that using two parameters? 17 00:00:51,009 --> 00:00:53,640 So what we want to do, is you can just visualize it, a 18 00:00:53,640 --> 00:00:56,170 doughnut, if you were to draw some axes here. 19 00:00:56,170 --> 00:00:57,230 So that's our doughut. 20 00:00:57,229 --> 00:00:59,309 Let me draw some axes. 21 00:00:59,310 --> 00:01:03,100 So let's say I have the z-axis that goes straight up and down. 22 00:01:03,100 --> 00:01:05,030 So we have drawn it here, the doughnut's a little at a 23 00:01:05,030 --> 00:01:07,430 tilt, so the z-axis, I'll tilt it a little bit. 24 00:01:07,430 --> 00:01:11,580 So our z-axis goes straight through the center 25 00:01:11,579 --> 00:01:13,649 of the doughnut. 26 00:01:13,650 --> 00:01:17,880 So that right there, this is going to be an exercise in 27 00:01:17,879 --> 00:01:19,810 drawing more than anything else. 28 00:01:19,810 --> 00:01:23,829 So that is my z-axis, and then you can imagine the z-axis goes 29 00:01:23,829 --> 00:01:28,019 from there, and then this coming out of here 30 00:01:28,019 --> 00:01:30,469 will be my x-axis. 31 00:01:30,469 --> 00:01:35,890 That right there is my x-axis, and then maybe my y-axis 32 00:01:35,890 --> 00:01:38,189 comes out like that. 33 00:01:38,189 --> 00:01:40,209 And the whole reason why I drew it this way, is that if you 34 00:01:40,209 --> 00:01:42,859 imagine the cross section of this doughnut, I'll draw a 35 00:01:42,859 --> 00:01:45,409 little bit neater, but the cross section of this doughnut 36 00:01:45,409 --> 00:01:48,709 in the x-z axis, is going to look something like this. 37 00:01:48,709 --> 00:01:51,819 If I were to slice it in the x-z axis, it would look 38 00:01:51,819 --> 00:01:52,569 something like that. 39 00:01:52,569 --> 00:01:57,579 40 00:01:57,579 --> 00:01:58,579 That would be the slice. 41 00:01:58,579 --> 00:02:00,510 It would trace out, and we're thinking about not 42 00:02:00,510 --> 00:02:02,570 a full doughnut, just the surface of a doughnut. 43 00:02:02,569 --> 00:02:04,439 So it would trace out a circle like this. 44 00:02:04,439 --> 00:02:07,569 If you were to cut the doughnut in the positive z-y axis, it's 45 00:02:07,569 --> 00:02:13,620 going to trace out a circle that looks something 46 00:02:13,620 --> 00:02:14,599 like that, right there. 47 00:02:14,599 --> 00:02:17,539 And if you go out here, you're going to get bunch of circles. 48 00:02:17,539 --> 00:02:20,150 So if you think about it, it's a bunch of circles rotated 49 00:02:20,150 --> 00:02:21,950 around the z-axis. 50 00:02:21,949 --> 00:02:25,939 If you think of it that way, it'll give us some good 51 00:02:25,939 --> 00:02:29,189 intuition for the best way to parameterize this thing. 52 00:02:29,189 --> 00:02:31,900 So let's do it that way. 53 00:02:31,900 --> 00:02:34,030 Let's start off with just the z-y axis. 54 00:02:34,030 --> 00:02:36,689 I'll draw it a little bit neater than I've done here. 55 00:02:36,689 --> 00:02:45,240 So that is the z-axis, and that is y, just like that. 56 00:02:45,240 --> 00:02:48,629 And let's say that the center of these circles, let's say it 57 00:02:48,629 --> 00:02:53,349 lies on, you know, it can lie, when you cross the z-y axis, 58 00:02:53,349 --> 00:02:55,219 the center sits on the y-axis. 59 00:02:55,219 --> 00:02:56,770 I didn't draw it that neatly here, but I think 60 00:02:56,770 --> 00:02:58,730 you can visualize. 61 00:02:58,729 --> 00:03:01,069 So it sits right there on the y-axis. 62 00:03:01,069 --> 00:03:06,199 And let's say that it is a distance b away from the center 63 00:03:06,199 --> 00:03:10,329 of the torus, or from the z-axis it's a distance of b. 64 00:03:10,330 --> 00:03:13,170 It's always going to be a distance of b. 65 00:03:13,169 --> 00:03:15,509 It's always, if you imagine the top of the doughnut, let me 66 00:03:15,509 --> 00:03:17,419 draw the top of the doughnut. 67 00:03:17,419 --> 00:03:21,629 If you're looking down on a doughnut, let me draw a 68 00:03:21,629 --> 00:03:25,419 doughnut right here, if you're looking down on a doughnut, it 69 00:03:25,419 --> 00:03:27,709 just looks something like that. 70 00:03:27,710 --> 00:03:30,760 The z-axis is just going to be popping straight out. 71 00:03:30,759 --> 00:03:36,509 The x-axis would come down like this, and then the y-axis would 72 00:03:36,509 --> 00:03:39,370 go to the right, like that. 73 00:03:39,370 --> 00:03:41,210 So you can imagine, I'm just flying above this. 74 00:03:41,210 --> 00:03:43,670 I'm sitting on the z-axis looking down at the doughnut. 75 00:03:43,669 --> 00:03:46,000 It will look just like this. 76 00:03:46,000 --> 00:03:49,719 And if you imagine the cross section, this circle right 77 00:03:49,719 --> 00:03:57,050 here, the top part of the circle if you're looking down, 78 00:03:57,050 --> 00:03:59,450 would look just like that. 79 00:03:59,449 --> 00:04:04,679 And this distance b is a distance from the z-axis 80 00:04:04,680 --> 00:04:06,099 to the center of each of these circles. 81 00:04:06,099 --> 00:04:09,919 So this distance, let me draw it in the same color, from the 82 00:04:09,919 --> 00:04:15,280 center to the center of these circles, that is going to be b. 83 00:04:15,280 --> 00:04:17,000 It's just going to keep going to the center 84 00:04:17,000 --> 00:04:18,670 of the circles, b. 85 00:04:18,670 --> 00:04:21,840 That's going to be b, that's going to be b. 86 00:04:21,839 --> 00:04:23,369 That's going to be b. 87 00:04:23,370 --> 00:04:26,590 From the center of our torus to the center of our circle 88 00:04:26,589 --> 00:04:29,810 that defines the torus, it's a distance of b. 89 00:04:29,810 --> 00:04:35,379 So this distance right here, that distance right there is b. 90 00:04:35,379 --> 00:04:38,779 And from b, we can imagine we have a radius. 91 00:04:38,779 --> 00:04:40,569 A radius of length a. 92 00:04:40,569 --> 00:04:42,860 So these circles have radius of length a. 93 00:04:42,860 --> 00:04:46,389 So this distance right here is a, this distance right here is 94 00:04:46,389 --> 00:04:49,519 a, this distance right there is a, that distance 95 00:04:49,519 --> 00:04:50,250 right there is a. 96 00:04:50,250 --> 00:04:51,910 If I were look at these circles, these circles 97 00:04:51,910 --> 00:04:53,650 have radius a. 98 00:04:53,649 --> 00:04:55,979 And what we're going to do is have two parameters. 99 00:04:55,980 --> 00:05:01,710 One is the angle that this radius makes with the x-z 100 00:05:01,709 --> 00:05:05,519 plane, so you can imagine the x-axis coming out. 101 00:05:05,519 --> 00:05:08,039 Let me do that in the same color. 102 00:05:08,040 --> 00:05:11,750 You can imagine the x-axis coming out here. 103 00:05:11,750 --> 00:05:13,569 So this is the x-z plane. 104 00:05:13,569 --> 00:05:17,480 So one parameter is going to be the angle between our 105 00:05:17,480 --> 00:05:19,840 radius and the x-z plane. 106 00:05:19,839 --> 00:05:24,669 We're going to call that angle, or that parameter, we're 107 00:05:24,670 --> 00:05:26,550 going to call that s. 108 00:05:26,550 --> 00:05:33,341 And so as s goes between 0 and 2 pi, as s goes between 0 in 2 109 00:05:33,341 --> 00:05:37,189 pi, when the 0 is just going to be at this point right here, 110 00:05:37,189 --> 00:05:40,959 and then as it goes to 2 pi, you're going to trace out a 111 00:05:40,959 --> 00:05:43,489 circle that looks just like that. 112 00:05:43,490 --> 00:05:45,069 Now, we only have one parameter. 113 00:05:45,069 --> 00:05:47,949 What we want to do is then spin this circle around. 114 00:05:47,949 --> 00:05:51,550 What I just drew is that circle right there. 115 00:05:51,550 --> 00:05:56,060 What we want to do is spin the entire circle around. 116 00:05:56,060 --> 00:06:00,209 So let's define another parameter. 117 00:06:00,209 --> 00:06:02,449 We'll call this one t, and I'll take the top view again. 118 00:06:02,449 --> 00:06:03,699 This one's getting a little bit messy. 119 00:06:03,699 --> 00:06:05,469 Let me draw another top view. 120 00:06:05,470 --> 00:06:07,700 As you can see, this is all about visualization. 121 00:06:07,699 --> 00:06:13,909 So let's say this is my x-axis, that is my y-axis. 122 00:06:13,910 --> 00:06:17,730 And we said we started here on the z-y plane. 123 00:06:17,730 --> 00:06:22,580 We are b away from the z-axis, so that distance is b. 124 00:06:22,579 --> 00:06:25,109 In this diagram, the z-axis is just popping out at us. 125 00:06:25,110 --> 00:06:26,160 It's popping out of the page. 126 00:06:26,160 --> 00:06:27,490 We're looking straight down. 127 00:06:27,490 --> 00:06:29,540 It's just like the same view as right there. 128 00:06:29,540 --> 00:06:35,170 And what I just drew, when s is equal to 0 radians, we're going 129 00:06:35,170 --> 00:06:37,860 to be out here, exactly one radius further 130 00:06:37,860 --> 00:06:38,699 along the y-axis. 131 00:06:38,699 --> 00:06:40,439 And then we're going to rotate. 132 00:06:40,439 --> 00:06:42,230 As we rotate around, we're going to rotate and then 133 00:06:42,230 --> 00:06:43,700 come all the way over here. 134 00:06:43,699 --> 00:06:45,759 That's when we're right over there, and then come back down. 135 00:06:45,759 --> 00:06:47,670 So if you looked on the top of the circle, it's 136 00:06:47,670 --> 00:06:49,410 going to look like that. 137 00:06:49,410 --> 00:06:50,910 Now, to make the doughnut, we're going to have to 138 00:06:50,910 --> 00:06:55,010 rotate this whole setup around the z-axis. 139 00:06:55,009 --> 00:06:56,500 Remember, the z-axis is popping out. 140 00:06:56,500 --> 00:06:58,920 It's looking straight up at us. 141 00:06:58,920 --> 00:07:01,340 It's coming out of your video screen. 142 00:07:01,339 --> 00:07:04,819 Now to rotate it, we're going to rotate this 143 00:07:04,819 --> 00:07:09,839 circle around the z-axis. 144 00:07:09,839 --> 00:07:12,489 And to do that, we'll define a parameter that tells us how 145 00:07:12,490 --> 00:07:16,210 much we have rotated it. 146 00:07:16,209 --> 00:07:18,939 So this is when we've rotated 0 radians. 147 00:07:18,939 --> 00:07:23,550 At some point, we're going to be over here, and we would 148 00:07:23,550 --> 00:07:27,490 have rotated it, this is b as well, and our circle is going 149 00:07:27,490 --> 00:07:29,389 to be looking like this. 150 00:07:29,389 --> 00:07:33,639 This is maybe this point on our doughnut, right there. 151 00:07:33,639 --> 00:07:35,389 At that point, we would have rotated it, 152 00:07:35,389 --> 00:07:37,229 let's say p radians. 153 00:07:37,230 --> 00:07:40,879 So this parameter of how much have we rotated around the 154 00:07:40,879 --> 00:07:44,240 z-axis, how much have we gone around that way, we're 155 00:07:44,240 --> 00:07:45,560 going to call that t. 156 00:07:45,560 --> 00:07:52,899 And t is also going to vary between 0 and 2 pi. 157 00:07:52,899 --> 00:07:53,959 And I want to make this very clear. 158 00:07:53,959 --> 00:08:00,829 Let's actually draw the domain that we're mapping from to 159 00:08:00,829 --> 00:08:04,689 our surface, so that we understand this fully. 160 00:08:04,689 --> 00:08:07,389 So let me draw some, and then we'll talk about how we can 161 00:08:07,389 --> 00:08:10,849 actually parameterize that into a position 162 00:08:10,850 --> 00:08:13,030 vector-valued function. 163 00:08:13,029 --> 00:08:21,029 So right here, let's call that the t-axis. 164 00:08:21,029 --> 00:08:23,329 That's, remember, how much we're rotated around 165 00:08:23,329 --> 00:08:25,009 the z-axis right there. 166 00:08:25,009 --> 00:08:31,610 And let's call this down here our s-axis. 167 00:08:31,610 --> 00:08:35,409 And I think this will help things out a good bit. 168 00:08:35,409 --> 00:08:39,179 So when s is equal to 0, and we vary just t, so they're 169 00:08:39,179 --> 00:08:41,269 both going to vary between 0 and 2 pi. 170 00:08:41,269 --> 00:08:46,189 So this right here is 0, this right here is 2 pi. 171 00:08:46,190 --> 00:08:47,540 Let me do some things in between. 172 00:08:47,539 --> 00:08:52,839 This is pi, this would be pi over 2 obviously, pi over 2, 173 00:08:52,840 --> 00:08:55,250 this would be 3 pi over 4. 174 00:08:55,250 --> 00:08:57,049 You do the same thing on the p-axis. 175 00:08:57,049 --> 00:08:59,549 It's going to go up to 2 pi. 176 00:08:59,549 --> 00:09:00,559 Let's do that. 177 00:09:00,559 --> 00:09:03,819 So we're going to go up to 2 pi. 178 00:09:03,820 --> 00:09:06,060 I really want you to visualize this, because then the 179 00:09:06,059 --> 00:09:10,189 parameterization, I think, will be fairly straightforward. 180 00:09:10,190 --> 00:09:16,240 So that's 2 pi, this is pi, this is pi over 2, and 181 00:09:16,240 --> 00:09:20,889 then this is 3 pi over 4. 182 00:09:20,889 --> 00:09:23,929 So let's think about what it looks like if you just hold s 183 00:09:23,929 --> 00:09:28,769 constant at 0, and we just vary t between zero and 2 pi. 184 00:09:28,769 --> 00:09:32,870 And let me do that in magenta, right here. 185 00:09:32,870 --> 00:09:35,480 So we're holding s constant, and we're just varying 186 00:09:35,480 --> 00:09:37,250 the parameter 2 pi. 187 00:09:37,250 --> 00:09:39,799 So this, if you think about it, should just form a curve in 188 00:09:39,799 --> 00:09:41,029 three dimensions, not a surface. 189 00:09:41,029 --> 00:09:43,500 Because we're only varying one parameter right here. 190 00:09:43,500 --> 00:09:45,370 So let's think about what this is. 191 00:09:45,370 --> 00:09:51,360 Remember, s is, let me draw my axes. 192 00:09:51,360 --> 00:09:58,330 So that is my x-axis, that is my y-axis, and then 193 00:09:58,330 --> 00:10:01,970 this is my, I'm getting messier and messier. 194 00:10:01,970 --> 00:10:05,629 That is my z-axis, right-- actually, let me draw it a 195 00:10:05,629 --> 00:10:06,860 little bit bigger than that. 196 00:10:06,860 --> 00:10:10,570 I think it will help all of our visualizations. 197 00:10:10,570 --> 00:10:11,080 All right. 198 00:10:11,080 --> 00:10:20,210 So this is my x-axis, that is my y-axis, and then my z-axis 199 00:10:20,210 --> 00:10:22,450 goes up like that. z-axis. 200 00:10:22,450 --> 00:10:25,360 Now remember, when s is equal to 0, that means we haven't 201 00:10:25,360 --> 00:10:29,550 rotated around this circle at all. 202 00:10:29,549 --> 00:10:31,299 That means we're out here. 203 00:10:31,299 --> 00:10:34,829 We're going to be b away, and then a away again. 204 00:10:34,830 --> 00:10:35,050 Right? 205 00:10:35,049 --> 00:10:36,990 We haven't rotated around this at all. 206 00:10:36,990 --> 00:10:39,629 We're setting s as equal to zero. 207 00:10:39,629 --> 00:10:43,649 So essentially, we're going to be b away, so this is going to 208 00:10:43,649 --> 00:10:46,799 be a distance of b away, and then we're going to 209 00:10:46,799 --> 00:10:48,479 be another a away. 210 00:10:48,480 --> 00:10:50,310 The b is the center of the circle, and then we're going 211 00:10:50,309 --> 00:10:51,299 to be another a away. 212 00:10:51,299 --> 00:10:52,549 We're going to be right over there. 213 00:10:52,549 --> 00:10:54,729 So this is a plus b away. 214 00:10:54,730 --> 00:10:56,129 And then we're going to vary t. 215 00:10:56,129 --> 00:10:59,870 Remember, t was how much we've gone around the z-axis. 216 00:10:59,870 --> 00:11:01,669 These were top views over here. 217 00:11:01,669 --> 00:11:07,639 So this line right here, in our s-t domain, we can say, when we 218 00:11:07,639 --> 00:11:11,939 map it, or parameterize it, it'll correspond to the curve 219 00:11:11,940 --> 00:11:13,980 that's essentially the outer edge of are doughnut. 220 00:11:13,980 --> 00:11:16,690 If this is the top view of the doughnut, it will be the outer 221 00:11:16,690 --> 00:11:18,800 edge of the doughnut, just like that. 222 00:11:18,799 --> 00:11:20,689 So let me draw the outer edge. 223 00:11:20,690 --> 00:11:22,620 And to do that a little bit better, let me draw the axes 224 00:11:22,620 --> 00:11:25,440 in both the positive and the negative domain. 225 00:11:25,440 --> 00:11:29,140 It might make my graph a little bit easier to visualize things. 226 00:11:29,139 --> 00:11:33,819 Positive and negative domain, this is negative z right there. 227 00:11:33,820 --> 00:11:37,480 So this line in our t-s plane, I guess we could say, this 228 00:11:37,480 --> 00:11:42,470 magenta line, we hold s at 0 radians and we increase t, this 229 00:11:42,470 --> 00:11:47,750 is t is zero, this is t is equal to 2 pi, that's t is 230 00:11:47,750 --> 00:11:53,289 equal to pi, this is t is equal to 3 pi over 2, all the way 231 00:11:53,289 --> 00:11:55,779 back to t is equal to 2 pi. 232 00:11:55,779 --> 00:12:00,470 This line corresponds to that line, as we rotate, as we 233 00:12:00,470 --> 00:12:04,889 increase t and hold s constant at 0. 234 00:12:04,889 --> 00:12:06,899 Now let's do another point. 235 00:12:06,899 --> 00:12:11,750 Let's say when s is at pi, right, remember, when s is 236 00:12:11,750 --> 00:12:14,659 at pi, we've gone exactly, pi is 180 degrees. 237 00:12:14,659 --> 00:12:17,429 When s is at pi, we've gone exactly 180 degrees 238 00:12:17,429 --> 00:12:19,339 around the circle, around each of these circles. 239 00:12:19,340 --> 00:12:20,810 So we're right over there. 240 00:12:20,809 --> 00:12:23,519 And now let's hold it constant at pi, and then rotate it 241 00:12:23,519 --> 00:12:24,710 around to form our doughnut. 242 00:12:24,710 --> 00:12:27,540 So we're going to form the inside of our doughnut. 243 00:12:27,539 --> 00:12:32,120 So when s is at pie, and we're going to take t from 0, so when 244 00:12:32,120 --> 00:12:35,620 s is pi and t is 0, we're going to be, this was the center of 245 00:12:35,620 --> 00:12:37,409 our circle, we're going to be a below that. 246 00:12:37,409 --> 00:12:39,929 We're going to be right over there. 247 00:12:39,929 --> 00:12:47,429 And then as we vary, as we increase t, so as we move up 248 00:12:47,429 --> 00:12:54,620 along, holding s at pi, and we increase t, we're going to 249 00:12:54,620 --> 00:13:02,070 trace out the inside of our doughnut, that will look 250 00:13:02,070 --> 00:13:04,170 something like that. 251 00:13:04,169 --> 00:13:06,339 That was my best shot at drawing it. 252 00:13:06,340 --> 00:13:08,050 And then we can do that multiple times. 253 00:13:08,049 --> 00:13:11,659 When s is pi over 2, I want to do multiple different colors, 254 00:13:11,659 --> 00:13:15,959 when s is pi over 2, we've rotated up here exactly 255 00:13:15,960 --> 00:13:16,980 90 degrees, right? 256 00:13:16,980 --> 00:13:18,899 Pi over 2 is 90 degrees at this point. 257 00:13:18,899 --> 00:13:21,500 And then if we vary t, we're essentially tracing out the 258 00:13:21,500 --> 00:13:23,840 top of the doughnut, right? 259 00:13:23,840 --> 00:13:25,110 So let me make sure I draw it . 260 00:13:25,110 --> 00:13:26,990 So the cross section, the top of the doughnut, we're going 261 00:13:26,990 --> 00:13:30,690 to start off right over here. 262 00:13:30,690 --> 00:13:34,910 So when s is pi over 2, and you vary it right, and then you 263 00:13:34,909 --> 00:13:39,100 very t, I'm having trouble drawing straight lines. 264 00:13:39,100 --> 00:13:42,500 And then you vary t, it's going to look like this. 265 00:13:42,500 --> 00:13:44,980 That's the top of that circle right there. 266 00:13:44,980 --> 00:13:47,420 The top of this circle is going to be right there. 267 00:13:47,419 --> 00:13:50,490 The top of this circle is going to be right over there. 268 00:13:50,490 --> 00:13:53,000 Top of the circle is going to be right over there. 269 00:13:53,000 --> 00:13:54,299 So then I just connect the dots. 270 00:13:54,299 --> 00:13:59,469 It's going to look something like that. 271 00:13:59,470 --> 00:14:00,769 That is the top of our doughnut. 272 00:14:00,769 --> 00:14:02,750 If I was doing this top view, it would be the top of the 273 00:14:02,750 --> 00:14:04,100 doughnut, just like that. 274 00:14:04,100 --> 00:14:06,750 And if I wanted to do the bottom of the doughnut, just to 275 00:14:06,750 --> 00:14:10,529 make the picture clear, if I were to make the bottom of the 276 00:14:10,529 --> 00:14:13,350 doughnut, the bottom of the doughnut would be-- see, if I 277 00:14:13,350 --> 00:14:17,210 take s as 3 pi over 4 and I vary t, that's the bottoms 278 00:14:17,210 --> 00:14:17,850 of our doughnuts. 279 00:14:17,850 --> 00:14:21,870 So let me draw the circle, it's right there, the circle is 280 00:14:21,870 --> 00:14:24,399 right there, you wouldn't be able to see the whole thing 281 00:14:24,399 --> 00:14:26,949 if this wasn't transparent. 282 00:14:26,950 --> 00:14:30,660 So you'd be tracing out the bottom of the doughnut, 283 00:14:30,659 --> 00:14:31,350 just like that. 284 00:14:31,350 --> 00:14:33,860 I know this graph is becoming a little confusing, but 285 00:14:33,860 --> 00:14:34,860 hopefully you get the idea. 286 00:14:34,860 --> 00:14:38,470 When s is 2 pi again, you're going to be back to the outside 287 00:14:38,470 --> 00:14:40,220 of the doughnut again. 288 00:14:40,220 --> 00:14:43,460 That's also going to be in purple. 289 00:14:43,460 --> 00:14:46,480 So that's what happens when we hold the s constant at certain 290 00:14:46,480 --> 00:14:48,730 values and vary the t. 291 00:14:48,730 --> 00:14:50,779 Now let's do the opposite. 292 00:14:50,779 --> 00:14:55,139 What happens if we hold t at 0, and we very the s? 293 00:14:55,139 --> 00:15:00,509 294 00:15:00,509 --> 00:15:03,610 So t is 0, that means we haven't rotated at all yet. 295 00:15:03,610 --> 00:15:05,159 So we're in the z-y plane. 296 00:15:05,159 --> 00:15:16,689 So t is 0, and s will start at 0, and it'll go to pi over 2, 297 00:15:16,690 --> 00:15:18,180 that's that point over there. 298 00:15:18,179 --> 00:15:20,669 Then it'll go to pi. 299 00:15:20,669 --> 00:15:23,299 This point is the same thing as that point. 300 00:15:23,299 --> 00:15:28,019 Then it will go to 3 pi over 4, then it'll come back 301 00:15:28,019 --> 00:15:29,549 all the way to 2 pi. 302 00:15:29,549 --> 00:15:34,079 So this line corresponds to this circle, right there. 303 00:15:34,080 --> 00:15:43,270 We could keep doing these if we pick when t is pi -- let me use 304 00:15:43,269 --> 00:15:46,289 a different color, that's not different enough. 305 00:15:46,289 --> 00:15:50,870 When t is pi over 2, just like that. 306 00:15:50,870 --> 00:15:53,980 We would have rotated around the z-axis 90 degrees, 307 00:15:53,980 --> 00:15:55,279 so now we're over here. 308 00:15:55,279 --> 00:15:58,709 And now when we vary s, s will start off over here, and 309 00:15:58,710 --> 00:16:02,330 it'll go all the way around like that. 310 00:16:02,330 --> 00:16:05,025 So this line corresponds to that circle. 311 00:16:05,024 --> 00:16:08,179 We could keep doing it like this. 312 00:16:08,179 --> 00:16:11,609 When t is equal to pi, that means we've got all the way 313 00:16:11,610 --> 00:16:14,669 around the circle like that, and now when we vary s from 0 314 00:16:14,669 --> 00:16:18,469 to pi over 2, we're going to start all the way over here, 315 00:16:18,470 --> 00:16:20,590 and then we're going to vary, all the way, we're going to go 316 00:16:20,590 --> 00:16:24,430 down and hit all those contours that we talked about before, 317 00:16:24,429 --> 00:16:26,919 and I'll do one more, just to kind of make this, 318 00:16:26,919 --> 00:16:29,509 the scaffold, clear. 319 00:16:29,509 --> 00:16:31,539 This dark purple, hopefully you can see it. 320 00:16:31,539 --> 00:16:34,149 When t is 3 pi over 4, we've rotated all the way. 321 00:16:34,149 --> 00:16:38,069 So we're in the x-z plane. 322 00:16:38,070 --> 00:16:41,900 And then when you vary s, s will start off over here, and 323 00:16:41,899 --> 00:16:48,459 as you increase s, you're going to go around the circle, around 324 00:16:48,460 --> 00:16:50,950 the circle, just like that. 325 00:16:50,950 --> 00:16:54,220 And of course, when you get all the way back full circle, t 326 00:16:54,220 --> 00:16:55,810 over pi over 2, that's the same thing. 327 00:16:55,809 --> 00:16:57,169 You're back over here again. 328 00:16:57,169 --> 00:17:01,620 So this is going to be, we can even shade it the same color. 329 00:17:01,620 --> 00:17:03,000 And hopefully you're getting a sense now of 330 00:17:03,000 --> 00:17:04,130 the parameterization. 331 00:17:04,130 --> 00:17:05,450 I haven't done any math yet. 332 00:17:05,450 --> 00:17:07,450 I haven't actually showed you how to mathematically represent 333 00:17:07,450 --> 00:17:10,360 it as a vector value function, but hopefully you're getting 334 00:17:10,359 --> 00:17:15,669 a sense of what it means to parameterize by two parameters. 335 00:17:15,670 --> 00:17:23,080 And just to get an idea of what these areas on our s-t plane 336 00:17:23,079 --> 00:17:30,349 correspond to onto this surface, in I guess you could 337 00:17:30,349 --> 00:17:34,639 say, in R3, this little square right here, let's see 338 00:17:34,640 --> 00:17:36,300 what's bounded by. 339 00:17:36,299 --> 00:17:38,899 This little square, I want to make sure I picked a square 340 00:17:38,900 --> 00:17:41,100 that I can draw neatly. 341 00:17:41,099 --> 00:17:46,754 So this square right here, that is between, when you look at t, 342 00:17:46,755 --> 00:17:50,070 it's between 0 and pi over 2. 343 00:17:50,069 --> 00:17:55,339 344 00:17:55,339 --> 00:17:58,740 And s is between 0 and pi over 2. 345 00:17:58,740 --> 00:18:04,120 So this right here is this part of our torus. 346 00:18:04,119 --> 00:18:07,079 347 00:18:07,079 --> 00:18:11,839 If you're looking at it from the top, it would look 348 00:18:11,839 --> 00:18:13,389 like that, right there. 349 00:18:13,390 --> 00:18:16,020 You can imagine, we've transformed this square. 350 00:18:16,019 --> 00:18:18,559 I haven't even shown you how to do it mathematically yet. 351 00:18:18,559 --> 00:18:22,129 But we've transformed this square to this 352 00:18:22,130 --> 00:18:24,570 part of the doughnut. 353 00:18:24,569 --> 00:18:27,339 Now, I think we've done about as much as I can do on 354 00:18:27,339 --> 00:18:28,529 the visualization side. 355 00:18:28,529 --> 00:18:30,490 I'll stop this video here. 356 00:18:30,490 --> 00:18:31,970 In the next video, we're going to actually talk about, how 357 00:18:31,970 --> 00:18:35,289 do we actually parameterize using these two parameters? 358 00:18:35,289 --> 00:18:39,379 Remember, s takes around each of these circles, and then t 359 00:18:39,380 --> 00:18:41,090 takes us around the z-axis. 360 00:18:41,089 --> 00:18:43,689 And if you take all of the combinations of s and t, you're 361 00:18:43,690 --> 00:18:46,049 going to have every point along the surface of this 362 00:18:46,049 --> 00:18:47,470 torus or this doughnut. 363 00:18:47,470 --> 00:18:50,759 How do you actually go from an s and a t that goes from 0 to 364 00:18:50,759 --> 00:18:56,029 2 pi, in both cases, and turn it into a three-dimensional 365 00:18:56,029 --> 00:19:00,129 position vector-valued function that would define this surface? 366 00:19:00,130 --> 00:19:02,280 We're going to do that in the next video.