Cardioid

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The red curve is a cardioid.

In geometry, the cardioid is an epicycloid with one cusp. That is, a cardioid is a curve that can be produced as the path (or locus) of a point on the circumference of a circle as that circle rolls around another fixed circle with the same radius.

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 only has one sharp point (or cusp). 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 figure in the Mandelbrot set is a cardioid.

Caustics can take the shape of cardioids. The caustic seen at the bottom of a coffee cup, for instance, may be a cardioid. The specific curve depends on the angle the light source makes relative to the bottom of the cup. The shape can also be a nephroid, which looks quite similar.

1 Equations
2 Graphs
3 Area
4 See also
5 References

[edit] Equations

Since the cardioid is an epicycloid with one cusp, it has parametric equations

$x(t) = 2r \left( \cos t - {1 \over 2} \cos 2 t \right),$

$y(t) = 2r \left( \sin t - {1 \over 2} \sin 2 t \right)$

where r is the radius of the circles which generate the curve, and the fixed circle is centered at the origin. The cusp is at (r,0).

The polar equation

$\rho(\theta) = 2r(1 - \cos \theta). \$

yields a cardiod with the same shape. It is the same curve as the cardioid given above, shifted to the left r units, so the cusp is at the origin.

For a proof, see cardioid proofs.

[edit] Graphs

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

[edit] Area

The area of a cardioid with polar equation

ρ(θ) = a (1 - cosθ)

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

See proof.

[edit] See also

Wittgenstein's rod
microphone - for a discussion of cardioid microphones

[edit] 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)

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Category: Curves