cardioid - Equations

The path of a point on a circle rolling around another circle of the same radius. The equation of a cardioid is best expressed in polar co-ordinates: r = a(1 + cos ?).

That is, a cardioid is a curve that can be produced as a locus — by tracing the path of a chosen point of a circle which rolls without slipping around another circle which is fixed but which has the same radius as the rolling circle.

The cardioid is also a special type of limaçon: it is the limaçon with one cusp. (The cusp is formed when the ratio of a to b in the equation is equal to one.)

The name comes from the heart shape of the curve (Greek kardioeides = kardia:heart + eidos:shape).

Equations

Since the cardioid is an epicycloid with one cusp, its parametric equations are

The same shape can be defined in polar coordinates by the equation