Cardioid

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The red curve is a cardioid.
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 be a nephroid, which looks quite similar.

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[edit] Equations

Since the cardioid is an epicycloid with one cusp, in cartesian coordinates 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 cardioid 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)

is

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

See proof.

[edit] See also

[edit] References