Cardioid
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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 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 ♥ 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.
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Equations
Since the cardioid is an epicycloid with one cusp, its parametric equations are
- <math> x(\theta) = \cos \theta + {1 \over 2} \cos 2 \theta, \qquad \qquad</math>
- <math> y(\theta) = \sin \theta + {1 \over 2} \sin 2 \theta. \qquad \qquad</math>
The same shape can be defined in polar coordinates by the equation
- <math> \rho(\theta) = 1 + \cos \theta. \ </math> 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
- <math> \rho(\theta) = a(1 - \cos \theta) </math>
is
- <math> A = {3\over 2} \pi a^2 </math>.
See proof.
See also
References
- Hearty Munching on Cardioids at cut-the-knot
- Xah Lee, Cardioid (1998) (This site provides a number of alternative constructions).
- Jan Wassenaar, Cardiod, (2005) in 862 two-dimensional mathematical curves.af:Kardioïed
ca:Cardioide de:Kardioide fr:Cardioïde ko:하트방정식 it:Cardioide ja:カージオイド pl:Kardioida pt:Cardióide ru:Кардиоида