The Cardioid

Introduction:

The cardioid (from the Latin cordis, meaning "the heart") is so named because it is heart shaped. It is a special case of the epicycloid in which the radius of the outer circle is the same as that of the inner circle, and a special case of the limaçon.

Definition:

In polar coordinates,

Geometrically, the cardioid is defined by the trace of a point on the circumference of a circle of radius a rolling without slipping on another circle of radius a. In the picture below, the highlighted point will trace the cardioid as the circle on the right rolls around the stationary circle on the left.

Properties:

Velocity and Acceleration:

The Tangent Vector:

Arc Length:

Curvature: