Cardioid

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Cardioid as a shadow of ring

Cardioid is closed curve with one cusp.

Contents

[edit] Definition

In geometry, the cardioid is an epicycloid with one cusp.

[edit] Construction

Rolling circle around another fixed circle produces cardioid (red curve)
Conformal mapping from circle to cardioid
  • epicycloid 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.
for\qquad n = \frac{1}{2}\,
  • it is a shadow of the ring on open book

[edit] Name

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.


[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(t) = 2r(1 - \cos t). \,

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

\rho (t) = a(1 - \cos t) \,

is

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

See proof.

[edit] Examples

[edit] Mandelbrot set

Cardioid im Mandelbrot set

There are many cardioids in Mandelbrot set [4]:

  • boundary of large central figure ( period 1 hyperbolic component) is a cardioid with equation :
 c = \frac{e^{it}}{2}  - \left (\frac{e^{it}}{2}\right )^2   \,
  • second largest cardioid is boundary of period 3 component on main antennae,
c = \left ( \frac{(P-1)\sqrt{27P^2-22P+23}}{6\sqrt{3}}-\frac{27P^2-36P+25}{54}\right ) ^{1/3}+
 \frac{3P+1}{9\left(\frac{ (P-1) \sqrt{27P^2-22P+23}}{6 \sqrt{3}} -\frac{27P^2-36P+25}{54} \right )^{1/3}} - \frac{2}{3} \,

where P = \frac{e^{it}}{2^3} \,

  • generealy every mini copy of Mandelbrot set contains one cardioid.

[edit] Caustics

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.

[edit] See also

[edit] Bibliography

  • Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. pp. 24–25. ISBN 0-14-011813-6. 

[edit] References

  1. ^ Weisstein, Eric W. "Inverse Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCurve.html
  2. ^ Weisstein, Eric W. "Parabola Inverse Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ParabolaInverseCurve.html
  3. ^ 3D-XplorMath \ Conformal Maps \ a*z^b+b*z
  4. ^ Boundary of hyperbolic components of Mandelbrot set

[edit] External links

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