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
(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.