The Cycloid

The cycloid is a special type of parametric curve that is traced out by a point on the circumference of the circle as it rolls along a straight line. The graph a cycloid looks like:

Parametric Equation for a Cycloid

First let's determine the center of the circle. For the x-coordinate, notice the arc formed as point P rolls along the x-axis is equal to the distance between the origin and the center of the circle (this is expanded on in the next section), and also notice that the y-coordinate of the circle does not change ever and stays at a length r. Thus we obtain that since the arc length is rΘ, then the center of the circle is C(rΘ, r)

x-Coordinates of a Cycloid

To determine the parametric equation for the cycloid, let's use the angle theta created by a perpendicular dropped from the center of the circle and the position of some point P which traces out the circle as theta increases. We will use theta because it varies like t for time would. The diagram below is a visual representation of how we will define the cycloid

First let's find a function x(t) to describe how the x-coordinate of the cycloid changes as theta varies. Note that point P starts off at the origin and moves away at a distance equal to that of the length of the segment OC. However, also notice that the arc of the circle formed by theta is also equal to the segment OC because it had rolled across the x-axis. Recall that the arc of a circle is:

(1)

\begin{align} s = r \theta \end{align}

We don't need to know an exact measurement of the radius, and theta varies. So simply:

(2)

\begin{align} OC = r \theta \end{align}

The length of OC is not what we are only looking for however. We need to subtract the length of the segment PC to get the length of OP = x. We can use trigonometry in this case:

(3)

\begin{align} \sin \theta = \frac{PC}{r} \\ r \sin \theta = PC \end{align}

Thus the coordinates of x as theta varies is:

(4)

\begin{align} OC - PC \\ = r \theta - r \sin \theta \\ = r(\theta - \sin \theta) \end{align}

y-Coordinates of a Cycloid

We will use similar techniques to determine the y-coordinate as theta varies. We will first acknowledge that the length of the dropped perpendicular from the center of the circle to the x-axis is equal to r, since it is just a radius of the circle. We now want to subtract the distance of the center to the y-coordinate. Once again, we can use trigonometric to get it's length to be rcosΘ. Thus, we obtain that the y-coordinate is equal to:

(5)

\begin{align} y = r - r \cos \theta \\ y = r(1 - \cos \theta) \end{align}

Analysis of the Cycloid

We learned that the cycloid can be defined by two parametric equations, namely:

(6)

\begin{align} x = r(\theta - \sin \theta) \quad , \quad y = r(1 - \cos \theta) \end{align}

Since the point tracing out the cycloid, P, starts at the x-axis and rolls from the x-axis, it makes sense that it takes the cycloid a distance of 2πr to intersect the x-axis again. Thus, one arch of the cycloid occurs after a distance of 2πr rolls of the cycle takes place.

Let's now analyze to see if these arches are semicircles. We know that the maximum height of the circle is going to be twice the radius, or 2r, which will also be the maximum height of the arch. We just defined the distance from the endpoint of one arch to the other endpoint to be 2πr. Clearly 2r ≠ 2πr/2 and 2r ≠ πr, thus, these arches are not semicircles.