Cycloid: The Curve Traced by a Rolling Wheel

A cycloid is the curve traced by a point on the circumference of a circle as it rolls along a straight line without slipping. The parametric equations are:

$x(t) = r(t - \sin t), \quad y(t) = r(1 - \cos t)$

where $r$ is the wheel radius and $t$ is the angle the wheel has rotated (in radians).

The cycloid has remarkable properties: it's the brachistochrone (the curve of fastest descent under gravity) and the tautochrone (objects released at any point on an inverted cycloid reach the bottom at the same time, regardless of starting position). Huygens used this property to design pendulum clocks.

Try it in the visualization

Watch the wheel roll along the ground. The red dot on the rim traces the cycloid arch. Adjust the wheel radius and see how the arch scales. Toggle "epicycloid" or "hypocycloid" to see what happens when the wheel rolls on the outside or inside of another circle.