Cardioid

In geometry, the cardioid 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 lima�on: 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 ♥ symbol, though, it doesn't have the sharp point at the bottom.

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. \$ (proof)

Graphs

Missing image
CardioidsLabeled.PNG
Image:CardioidsLabeled.PNG

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

External link

Alexander Bogomolny, Hearty Munching on Cardioids at Cut-the-Knot, (1998)
Xah Lee, Cardioid (1998) (This site provides a number of alternative constructions).

Cardioid

Equations

Graphs

External link

See also: Cardioid, Cardinal direction, Cardioid/Proofs, Circle, Curve, Cusp, Epicycloid, Equation, Fractal, Geometry