A cycloid is the curve traced by a point on a circle as the circle rolls along a line.

Roberval (1634) found the velocity vector (this is modern terminology) that describes both the tangent direction for a point on the cycloid, and the velocity of the point moving along the curve as the circle rolls at a constant rate.

See the lecture notes on the cycloid for the proof.

Technical details: use the "Roll" slider, or click the "Play" triangle in the lower left of the window to roll the circle.

Susan Addington, Created with GeoGebra