The Cycloid

Introduction:

The cycloid is a famous curve traced by a point on the edge of a rolling wheel.

Definition:

Geometrically, the cycloid is formed by tracing a point on the edge of a circle B of radius b that is concentric to a circle A of radius a, as circle A rolls without slipping along the x-axis. If the radius of B is less than that of A, then the cycloid is said to be curtate. If the radius of A is smaller, the cycloid is said to be prolate.

A curtate cycloid. A prolate cycloid.

 

Properties:

Velocity and Acceleration:

The Tangent Vector:

Arc Length:

Curvature: