Cycloids and Parametric Equations

by

Stacy Musgrave

The cycloid is the perfect example to show students why parametric curves are useful. To describe a cycloid without parametric curves is likely to cause a headache. With parametric equations, however, the task becomes quite doable.

Recall, a cycloid is the shape we obtain by tracing a point on a circle as the circle rolls along a straight line at a fixed speed over some period of time.

Now, let's actually find the equations for the parametric curves that would give a cycloid formed by rolling a circle of radius r, centered at (0,r), along the x-axis. We'll say the center of the circle is translated the distance the circle has rolled, so the center remains along the horizontal line y = r and the x-coordinate varies depending on time t. If we allow for clockwise rotation (rolling left to right along the x-axis), then we'll say the angle of rotation is -t, where t is the time allowed to pass. Note t is negative because we're going clockwise and t is positive for this demonstration. Then the distance the circle has traveled is the arc length of the circle subtended by angle -t. Namely, the circle has traveled a distance (r)(t).

If we suppose that the point on our circle started at the origin, we could calculate the effect this rotation has had on our point. Looking at the diagram below, we see that the new coordinates for the point on the circle are ( r(t - sin(t)) , r(1-cos(t)) ).

**Note the angle is marked -t to show the direction of rotation, but for calculations we use t.

Since the overall horizontal distance of the center of the circle is rt, the ending position for the x-coordinate of the point we care about is rt - a = rt - rsin(t) and the y-coordinate is r - b = r - rcos(t) as stated above.