Differential Geometry and Its Application - Exercise 1.1.13

Problem:

Suppose a circle of radius $a$ sits on the x-axis making contact at (0, 0). Let the circle roll along the positive x-axis. Show that the path $a$ followed by the point originally in contact with the x-axis is given by:

$\alpha(t) = (a(t - \sin t), a(1 - \cos t))$

where $t$ is the angle formed by the (new) point of contact with the axis, the center and the original point of contact. This curve is called a cycloid.

Solution;

Once we have the diagram, it is a simple matter to argue the formula:

As we can see, the point $p$ should have coordinate $at$ because the circle has rolled for $t$ radian. Now the base of the triangle is given by $a \sin t$, so the x coordinate is $at - a\sin t = a(t - \sin t)$.

For y coordinate, we see the height of the triangle is $a \cos t$, so the y-coordinate is $a(1 - \cos t)$.

So we have proved the parametric formula for the cycloid.