 Parametric functions arise when describing locations based on other locations. For example, we might have a point. On a rotating disk, that's moving. So, for example, maybe an ant is at the edge of a disk 2 meters in diameter. At t equals 0, the disk begins to rotate counterclockwise once every 10 seconds, while the center of the disk begins moving to the right at 5 meters per second. Let's find the parametric functions giving the position of the ant as a function of time t. So a useful idea here is take a small step. So here we'll let the disk move just a little bit. Now let's put down a set of coordinate axes where the center is the center of the disk. So suppose the disk travels for t seconds. The center will be at the position 5t0. The disk will have rotated t tenths of a rotation, and this corresponds to an angle of t tenths of 2 pi, or pi t fifths radians. So the line between the center of the disk and the ant makes an angle of pi t fifths radians with the x-axis. Now remember coordinates specify how to get to a position from the origin. So to get to the ant's position, you might go horizontally 5t to the center of the circle, and then backtrack to this position. This would put you in the right place horizontally. And remember since the circle itself is 2 meters in diameter, the radius is 1 meter, and so this length will be the cosine of pi t fifths. So the x-coordinate of the ant's position will be 5t minus cosine pi t fifths. Now to get to the ant's position, we have to go down this distance, and that's going to be sine of pi t fifths. So the y-coordinate of the ant's position will be negative sine of pi t fifths. Another famous curve is known as the cycloid. This is the path of a fixed point on a moving wheel as it rolls along the ground. Suppose the wheel is a meters in radius. Let's write the equation of the cycloid using the initial point of contact of the wheel and the ground. So again let's take a small step and let the wheel turn just a little bit. So again let's start with our wheel in the initial position and throw down a set of coordinate axes. This time we'll make our point be at the origin. We'll take a small step and roll the wheel a little bit. So suppose the wheel has made a rotation of theta. So that means the central angle has measured theta. Now because the wheel is rolling along the ground, that means the arc length has to equal the distance traveled. This means that it will have moved horizontally a distance of a theta, that's the arc length, and the point will have been rotated through an angle of theta. So again the coordinates of a point are a way of expressing how to get to that point from the origin. So to get to this point we might go out horizontally a distance of a theta, and then back a distance of a sine theta. Consequently our x coordinate would be a theta minus a sine theta. The vertical distance we might go up to the center of the circle and then back down a cosine theta. So the y coordinate would be a minus a cosine theta. Or let's consider the following. Given a point p and a line l, the controid is the curve whose points q satisfy the property that if pq passes through l at r, then rq is some fixed length k. Let's find the parametric equation of the controid where p is 1 meter below the line l and k is equal to 2, and they'll parameterize using theta, the angle of intersection of pq and l. So for convenience we'll assume that our line l is the x-axis and p is of the point 0 negative 1. So notice that as angle theta changes the location of q also changes. Drawing the line 2q and some perpendicularities. And remember we're assuming that rq has length 2. We see that qs, that's 2 sine theta, rs, that's 2 cosine theta, rt, that's the distance of p is below the axis, that's going to be 1, and pt will be cotangent theta. This means our x-coordinate will be cotangent theta plus 2 cosine theta, and our y-coordinate is 2 sine theta.