Cycloid

An interactive cycloid — the path traced by a point on the rim of a circle as it rolls along a straight line. Drag the sliders to change the circle's radius and roll it forward. Looking for a different curve? Browse the full list of interactive curves.

Cycloid

Drag the "t" slider to roll the circle forward and watch the traced point sweep out the cycloid; drag "a" to change the circle's radius.

Cycloid — The Details

A cycloid is the curve traced by a point fixed to the rim of a circle as that circle rolls, without slipping, along a straight line. Each arch of the cycloid spans a width equal to the circle's circumference and rises to a height equal to the circle's diameter. The cycloid has a long history in mathematics — Galileo named and studied it, and it later became famous as the solution to two classic problems: the brachistochrone (the shape of the fastest frictionless slide between two points under gravity) and the tautochrone (a ball released from any point on an inverted cycloid reaches the bottom in the same amount of time, no matter where it starts). For a circle of radius $a$ rolling along the x-axis with parameter $t$, its parametric equations are:

$x(t) = a(t - \sin t)$

$y(t) = a(1 - \cos t)$