Parametric Equations for a Cycloid

 Gayle Gilbert & Greg Schmidt

Let  be a point on a circle of radius   


Consider the curve, which is traced out by the point as the circle rolls along the -axis.  We will allow that our circle begins to trace the curve with the point  at the origin. 


Click here to see the animation in GSP.


Such a curve is called a cycloid.


Now, we can find the parametric equation fir the cycloid as follows:


Let the parameter be the angle of rotation of  for our given circle.  Note that  when the point  is at the origin.


Next consider the distance the circle has rolled from the origin after it has rotated through  radians, which is given by  











And so we can see that the center of the circle is given by .



Now, letting the coordinates of P be  we have that                 






Hence, we have




which gives us the parametric equations of the cycloid.