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.