Cardioid
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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 + \cos 2 \theta, \qquad \qquad]
- [ y(\theta) = \sin \theta + \sin 2 \theta. \qquad \qquad]
- [ \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 = \pi a^2 ].
See also
- Wittgenstein's rod
- microphone#Directionality
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.
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