Cardioid

In geometry, the cardioid, literally heart shape, is an epicycloid which has one and only one cusp. 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). Compared to the heart symbol (♥), though, a cardioid does not come to a sharp point. It is rather shaped more like the outline of the cross section of a plum.

The cardioid is an inverse transform of a parabola.

The large, central, black figure in a Mandelbrot set is a cardioid. This cardioid is surrounded by a fractal arrangement of circles.

Equations

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

x(\theta) = \cos \theta + {1 \over 2} \cos 2 \theta, \qquad \qquad

y(\theta) = \sin \theta + {1 \over 2} \sin 2 \theta. \qquad \qquad

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

\rho(\theta) = 1 + \cos \theta. \

For a proof, see cardioid proofs.

Graphs

Four graphs of cardioids oriented in the four cardinal directions, with their respective polar equations.

Area

The area of a cardioid which is congruent to

\rho(\theta) = a(1 - \cos \theta)

A = {3\over 2} \pi a^2

See proof.

References

Hearty Munching on Cardioids at cut-the-knot
Xah Lee, Cardioid (1998) (This site provides a number of alternative constructions).
Jan Wassenaar, Cardioid, (2005) in 862 two-dimensional mathematical curves.

Curves

Gourt Home::Gourt :: Cardioid

Equations

Graphs

Area

See also

References