A Cycloid in Parametric Form

by

D. Hembree

A cycloid is the locus of a point on a circle as the circle rolls along a line.Click here to open Geometer's Sketchpad and see an animation of a cycloid.

Parametric equations for motions in the plane are often easy to derive when the motion is based on a geometric relationship.
The following figure will aide in explaining some mathematics:

 
Circle C is tangent to a line at the origin S. As the circle rolls to the right, we wish to locate a point A on the circle by its coordinates (x,y).  When the circle has rolled slightly to the right, point A is now located as shown. If angle t is measured in radians, then length SG is the same as the length of arc AG, so G has coordinates (rt,0)  and C has coordinates (rt,r) In right triangle ABC, segment AB has length r sin(t) and CB has length r cos(t) so that the coordinates of point A are 
(SG-AB, GC-BC)
 which is 
(rt - r sin(t), r - r cos(t)) 

So parametric equations for a cycloid are x = rt - r sin(t) and y = r - r cos(t).

A Graphing Calculator image is shown below with r = 2. If you would like to play with the graph, click anywhere on the figure to open Graphing Calculator.

Extensions:  It is easy in Geometer's Sketchpad to look at the locus of a point outside a circle or inside a circle as the circle rolls along a line. Modify the GSP file at the top of the page to draw these cycloids. Write parametric equations for those situations, known as prolate or curtate cycloids, respectively. 
Return to D Hembree's  EMAT 6680 page
Email to  dhembree@coe.uga.edu