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Current time:0:00Total duration:16:32
Sal wrote "3pi/4" but meant "3pi/2".

Determining a position vector-valued function for a parametrization of two parameters

Video transcript

in the last video we started to talk about how to parameterize a torus or a doughnut shape and the two parameters were using and I spend a lot of time trying to visualize it because this is all about visualization I think this is really the hard thing to do here but the way we can parameterize the surface of our torus which is the surface of this doughnut is say hey let's take a point and let's take let's rotate it around a circle it could be any circle I picked a circle in the zy plane and how far it's gone around that circle will parameterize that by s and s can go between zero all the way to two pi and then we're going to rotate this circle around itself or I guess even better way to say we're going to rotate the circle around the z-axis and it's all at the center of the circles we're always going to keep a distance B away and so these were top views right there and then we defined our second parameter T which tells us how far the entire circle has rotated around the z-axis and those were those are two parameter definitions and then here we try to visualize what happens this is kind of the domain that our parameterization is going to be defined on s goes between zero and two pi so when T is zero we haven't rotated out of the zy plane s is it zero goes all the way to 2pi over there then when T goes to 2pi we've kind of moved moved our circle we've moved it along we've rotated around the z-axis a bit and then this line in our s T domain corresponds to that circle in three dimensions or in our XYZ space now given that hopefully we visualize it pretty well let's think about actually how to define a position vector-valued function that will that is essentially this parameterization so let's first do the Z because that's pretty straightforward so let's look at this view right here what's our Z going to be as a function so our X are wise and our Z should all be a function of s and T that's what it's all about we want to any position in space should be a function of picking a particular tea and a particular s and we saw that over here this point right here let me actually do that with a couple of points this point right there that corresponds to that point right there let me pick another one this point right here corresponds to this point right over there I could do a few more let me pick this point right here so s is still 0 that's going to be this outer edge way out over there I'll pick one more just to define this square this point right over here where we haven't rotated T at all but we've gone halfway or we've gone a quarter way around the circle is at that point right there so for any s and T we're mapping it to a point in X Y z space so our Z's our X's and our Y should all be a function of s and T so the first one to think about is just the Z and I think this will be pretty straight forward so Z as a function of s and T is going to equal what well if you take any circle remember s is how much we're going how much the angle between our radius and the XY plane so I can even draw it over here let me do it in another color it's um running out of colors so let's say that this is a radius right there that angle we said that is s so if I were to draw that circle out if I were to draw that circle out just like that we can do a little bit of trigonometry this is the angle is s we know the radius is a the radius of our circle we define that so the Z is just going to be the distance above or the distance above the XY plane it's going to be this distance right there and that's straightforward trigonometry that's going to be a I mean we could do sohcahtoa and all of that you might want to review the videos but the sine you can view it this way so this is if this is Z right there you could say that the sine of s sohcahtoa is the opposite of the hypotenuse is equal to Z over a multiplied both sides by a you have a sine of s is equal to Z that tells us how much above the XY plane we are that you know just some simple trigonometry so Z of s and T it's only going to be a function of s it's going to be a a times the sine of s not too bad now let's see if we can figure out what x and y are going to be remember Z doesn't matter it doesn't matter how much we've rotated around around the z axis what matters is how much we've rotated around the circle if s is 0 we're just going to be in the XY plane Z is going to be 0 if X is I'm sorry if s is PI over 2 or up here then we're going to be traveling around the top of the doughnut and we're going to be exactly a above the XY plane or z is going to be equal to a hopefully that makes reasonable sense to you now let's think about what happens what happens as we rotate around remember these two are top views we are looking down we're looking down on this doughnut so the center of each of these circles is B away from the origin or from mat from the origin or from the z-axis what we're rotating around always be away so our x coordinate or our x and y coordinate so if we go to the center of the circle here we're going to be be away and then this is going to be B away right over there so let's think about kind of where we are in the XY plane or how far the part of our when we're I guess you could imagine the if you were to project our point into the XY plane how far is that going to be from our origin well it's always going to be remember let's go back to this drawing here this might be the most instructive I'm not this is just one particular circle on the zy plane but could be any of them if this is the z axis over here that is the z axis this distance right here is always going to be B we know that for sure and so what is this distance going to be what is the distance what is this distance going to be where be to the center and then we're going to have some angle s and so depending on that angle s this distance onto I guess the XY plane you know for sitting on the XY plane how far are we from the z axis or the projection onto the XY plane or you can you know the the X or the Y position I'm saying it as many ways as possible I think you're visualizing it if Z is a sine of theta this distance right here this little shorter distance right here that's going to be a cosine of theta a cosine amount of theta of s right s is that angle right there this distance right here is going to be a cosine of s so if we talk about just straight distance from the origin along the XY plane our distance is always going to be it's always going to be B plus a cosine of s when a gets you know when s is out here then it's actually going to become a negative number and that makes sense because our distance is going to be less than B we're going to be at that point right there so if you look at these top views if you look at the top views over here no matter where we are you know that is B now let's say we've rotated a little bit that distance right here if you live along the XY plane that is always going to be B plus a cosine of s that's what that distance is to any given point we're depending on our SS and T's now as we rotate around if this if we're at a point here let's say we're at a point there and that point we already said is B plus a cosine of s away from the origin on the XY plane what are the X and y coordinates of that well this is a top down we're looking we're sitting on the z-axis looking straight down on the XY plane right now we're looking down on the doughnut so what is what are your x and y is going to be well you draw another right triangle right here you have another right triangle this angle right here is T eeee this distance right here is going to be this times the sine of our angle so this right here which is essentially our X this is going to be our x coordinate X as a function of s and T is going to be equal to the sine of T the sine of T T is our angle right there times this radius times we could write either way times B plus a cosine of s remember how far we are depends on how much around the circle we are right when we're over here we're much further away here we're exactly B away if you're looking only on the XY plane and then over here where B minus a away if we're looking if we're on the XY plane so that's X as a function of s and T and actually the way I defined it right here and it you know it this would our positive x-axis would actually go in this direction so this is X positive this is X in the negative direction I could have flipped the signs but hopefully you know this actually makes sense that that would be the positive x this is the negative X depends on whether using right-handed or left-handed coordinate system but hopefully that makes sense we're just saying okay what is this distance right here that is B plus a cosine of s we got that from this right here when we taking a view just kind of a cut of the torus that's how far we are in kind of the X Y direction at any point or kind of radially out without thinking about the height and then if you want the x coordinate you multiply it times the cosine or sorry you multiplied times the sine of T the way I've set it up here and the y coordinate the y coordinate is going to be this right here the way we've set up this triangle so Y as a function of s and T is going to be equal to the cosine of T cosine of T times this radius B plus a cosine of s and so our parameterization and you know just play with this triangle and hopefully it'll it'll make sense I mean if you say that this is our y coordinate right here you just do sohcahtoa cosine of t is equal to adjacent which is y right this is the angle right here over the hypotenuse over B plus a cosine of s multiply both sides of the equation times this you get Y you get Y of s of T is equal to cosine of T times this thing right there so let me copy and paste all of our takeaways let me copy it and then paste it and we're done with our parameterization we're done with our parameterization we could leave it just like this but if we want to define it as a if we want to represent it as a position vector-valued function we can define it like this find a nice color maybe pink so let's say our position vector function vector valued function is our it's going to be a function of two parameters s and T and it's going to be equal to its x value let me do that in the same color so it's going to be I'll do this part first B plus a cosine of s times sine of T and that's going to go in the x-direction so we'll say that's times I and in this case remember the way I defined it the positive x-direction is going to be here so the I unit vector will look like that I will look like will go in that direction the way I've defined things and then plus our Y value is going to be B plus a cosine of s times cosine of T in the Y unit vector direction remember the J unit vector will just go just like that that's our J unit vector and then finally we won't throw in the Z which was actually the most straightforward plus a sign of s times the K unit vector which is a unit vector in the Z direction so times the K unit vector and so you give me now any s and T within this domain right here any s and T within this domain right here and you give it put it into this position vector-valued function it'll give you the exact position vector that specifies the appropriate point on the torus so if you pick let's just make sure we understand what we're doing if you pick that point right there where s and T are both equal to PI over 2 and you might even want to go through the exercise take PI over 2 in all of these actually let's let's let's do it so in that case so when R of PI over 2 PI over 2 what do we get it's going to be B plus a times cosine of PI over 2 cosine of PI over 2 is 0 right cosine of 90 degrees so it's going to be B right this whole thing is going to be 0 times sine of PI over 2 sine of PI over 2 is just 1 so it's going to be B times I plus once again cosine of PI over 2 is 0 so this term right here is going to be B and then cosine of PI over 2 is 0 so it's going to be 0 J so that's going to be plus 0 J and then finally PI over 2 well you have a T there's no T here sine of PI over 2 is 1 so plus a times K plus a times K so there's actually no J direction so this is going to be equal to B times I plus a times K so the point that it specifies according to this parameterization or the vector expresses B times I plus a times K so B times I B times I will get this right out there and then a times K will get us right over there so the position vector being specified is right over there and just as we predicted that dot that point right there corresponds to that point just like that and of course I picked kind of points it was easy to calculate but this whole when you take every SN T in this domain right here you're going to transform it to this surface and this is the transformation right here and of course we have to specify that s is between we could write it multiple ways we could well we'll just write s is between two pi s is between two pi and 0 and we could also say T is between 2 pi and 0 and you could you know we're kind of overlapping one extra time at two pi so maybe we can get rid of one of these equal signs if you like although that won't necessarily area any if you're taking the surface area but hopefully this gives you at least a gut sense or more than a gut sense of how to parameterize these things and what we're even doing because it is going to be really important when we start talking about surface integrals and the hardest thing about doing all of this is just the visualization