The Cycloid

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slipping. The inverted cycloid (a cycloid rotated through 180°) is the curve of fastest descent under gravity. One arch of a cycloid generated by a circle of radius r has area = 3πr² The arc length S of one arch of the cycloid = 8r. credit to Guillermo Bautista