A cycloid is a curve traced by a point P on the rim of a wheel while the wheel rolls along a straight line. It is assumed that slippage does not occur during this motion.
To find the parametric equations for a cycloid, let the x-axis of the Cartesian coordinate system be the line on which the wheel rolls. Let the initial position of P(x,y) be the origin. We label the elements of the figure as follows:
C = the center of the wheel,
a = the radius of the wheel,
t = the clockwise angle (in radians) through which the segment CP has turned (The orignal position of CP is vertical.),
u = -pi - t, and
k = the directed arc with endpoints (N,P). (The central angle of k is the angle NCP.)
The arc length of k equals the product of the radius a and the measure of angle t. This value also is the distance along the x-axis which has been traveled by the wheel. So, at = |ON|.
Now, the x-coordinate of P(x,y) is |OM| = |ON| - |MN| = at - a(sin t) = a(t - sin t). We use the fact that |MN| = |PR| = a(sin u) = a(sin (-pi - t)) = a[(sin -pi)(cos t) - (cos -pi)(sin t)] = a(sin t). [Angle difference trigonometric identity]
The y-coordinate of P is |MP| = |NR| = |CN| + |CR| = a - a(cos t) = a(1 - cos t). We use the fact that |CR| = a(cos u) = a(cos (-pi - t)) = a[(cos -pi)(cos t) + (sin -pi)(sin t)] = a(-cos t) = -a(cos t).
Therefore, the parametric equations for a cycloid are
x = a(t - sin t)
y = a(1 - cos t).
Click on the illuminated values to view a Geometer's Sketchpad demonstration of the cycloids formed by wheels of radii 1 and 3^(1/2). These are motivated by letting the unit of measurement on the x-axis be radians, so that, as t travels along the axis, its position represents the clockwise angle through which the radius of the wheel has turned.
Reference: Calculus with Analytic Geometry (Sixth Edition), by Dale Varberg and Edwin J. Purcell