 Welcome back to our lecture series Math 1220 Calculus II for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misseldine. I want to continue our discussion about parametric curves that we had started in lecture 29. These examples can be, for most part, are taken from James Stewart's Calculus Textbook section 10.1 here about parametric curves. What I want to do in this video is talk about a very important parametric curve referred to as the cycloid. In the previous video, we saw a lot of parameterizations that came from trigonometry. This one will not be excluded from that one. It's a little bit more complicated, mind you, but it's still a pretty important one to talk about here, the cycloid. Imagine you have a circle, like this yellow one we see right here. This circle, we want to consider the curve that's traced out by a point on the circle. It's on the circumference of the circle, so this point P right here. We want to think of the point that's traced out as the circle rolls along a straight line, which we'll say is the x-axis right here. Now to help give a better illustration of this, I'm actually going to use a graphic, a gif from that taken courtesy of Wikipedia. You can find the link in the description below. So we have a circle with a fixed point, and so we're imagining if that point could trace a curve in space here, then what would the shape of that thing look like? And so this is what we mean by this cycloid, right? It looks like a cartwheel, maybe the wheel of a hand cart being pulled by a Mormon pioneer or something like that. As this cartwheel spins, imagine we tie a handkerchief to the cartwheel, and you're the 12 year old child who's given the task of counting how many times this thing spins around. Well, as you're going to hike around 12 miles in a given day, that's going to be a lot of rotations, and so you might get bored. And so you start imagining the path in the air of the handkerchief that we tied to the cartwheels and spins and spins and spins and spins, like you see happening right here. So we want to find an equation to describe this curve, this red curve you see in front of us, or at least we want to find parametric equations that can describe it. And so to do that, let's start to parameterize our cartwheel here, which we see in yellow, right? So just as it rolls along the x-axis key track of our point P, for convenience, we'll assume that P actually did pass through the origin where x and y-axis cross each other. Okay, so let's see. Suppose that the circle rolled some distance from the origin to some point P, right? And its coordinates will be given as x and y. Suppose x is measuring how far from the origin you are horizontally, and y measures how far above the x-axis you are vertically, alright? And we're going to let T be the point right here that is directly below the center of the circle at this given moment of time. So if we just take like a snapshot of our cartwheel and we look at where the handkerchief is on the wheel, we have this, we have the center of the circle, that will be where the axle is for the wheel. If we just look at the point directly above the ground, that will be my T, right there. So it's the intersection between the circle and the x-axis. And let Q be the intersection between the line TC and the horizontal line containing P. So if we take the line that's determined by the point C and T, that's Charlie and Tango there, and if we take the horizontal line that passes through the point P, we're going to call that point of intersection Q for Quebec there, alright? The radius of our circle, we're going to say is R, and the line PC, Papa Charlie there, with QT, so Quebec Tango there, it forms an angle. We're going to call that angle theta, and we're going to measure the angle PCT in radians for this discussion. And that way we can connect angles with arc length, which is going to be useful in this. So that's a description of all of the variables that are in play, the points and things like this. Again, this is just for one snapshot in time. So consider the line segment PC and CT. These are a radii of the circle, so their distances are both going to be R. So let's make a comment about that. So we know that the length of PC and the length of CT, these are both R because they're radii of the circle. Also, since the circle rolled along the x-axis, we know that the distance between O and T, so this distance right here, the distance from O to T, write that down, O to T, this is going to be the arc, the arc length of PT. That is, if we take the arc length from P to T right here, that's the same distance here because as it rolled from P to T, these parts were on the ground at some point. So we want to calculate this arc length there. Now this arc, the arc PT is associated to the angle PCT, right? And so by the usual arc length formula, this is going to equal R theta, as you see indicated in our diagram right there. Now let's do a little bit of trigonometry here. Consider the triangle PQC, right there it is, a right triangle with angle theta inside of it. Notice that this is, with respect to theta, this is the opposite side of the triangle, this is the hypotenuse, and therefore by a simple trigonometric argument, just so Catoa-type stuff, PQ will have length equal to R sine of theta. That's because PQ over PC is equal to sine and PC is equal to R, clear the denominators there. Likewise, this side, QC is the adjacent side of the triangle. So using the cosine relationship, we are going to get that QC, the length of the segment QC is equal to R cosine theta. All right, so we're gathering some good information right here. So now what we want to do is we want to describe the dimensions X and Y. So what we're trying to do, the location, we want to describe the location of these numbers X and Y. So if we start with X, X, it's actually just the distance along the line right here. Well, if we take the whole distance OT, right, we take the whole distance OT and we subtract from that the distance PQ. That's just going to be our X here, right? Because OT is this distance right here. If we subtract this distance right here, that just leaves us with X. That's nice and dandy. And since we have OT, OT is going to be R theta. And we also have PQ, that is R sine theta. We can describe the, we can describe the X with respect to this parameter theta right here. Now I noticed that there's a common factor of R right here. So if we factor that out, we're going to get X equals R times theta minus sine theta, like so. So we can describe parametrically X with respect to theta here. All right, what about Y? Y is going to be this distance right here. And so by a similar strategy, we're going to take the whole distance CT and we're going to subtract from it QC. So Y will equal CT minus QC, like so. And like we observed earlier, CT is a radius, so its length is R. And QC, as we can still see on the screen, is R cosine theta. And so if we were to factor out the R, we're going to get Y equals R times 1 minus cosine theta. And so this right here, these boxed equations, give us a parameterization of this cycloid, where we can describe the cycloid in terms of this parameter theta. And so as this, as your parameter, as your theta goes from 0 to 2 pi, that would give one complete rotation of the cartwheel, and therefore that forms one arch, that gives us one arch of our cycloid. As you go from, this is going to go from theta from 0 to 2 pi, right? Now in terms of distance, the horizontal distance along here would be up to 2 pi R, of course, because that's the radius of the circle. But one arch is parameterized from 0 to 2 pi. If we want to do the next arch, we'll go from 2 pi to 4 pi. If we want to do the next arch, we'll go from 4 pi to 6 pi. If we want to go back in time here, we could do this arch right here. This would be going from negative 2 pi to 0, things like that. So every multiple 2 pi would give us another arch of the cycloid. And so we have a very nice parameterization for our cycloid here. This is going to be a good friend that we're going to come back to several times in the next section.