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A cardioid is closed curve with one cusp.

Contents 1 Definition 2 Construction 3 Name 4 Equations 5 Graphs 6 Area 7 Examples 7.1 Mandelbrot set 7.2 Caustics 8 See also 9 Bibliography 10 References 10.1 External links

[edit] Definition

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

[edit] Construction

• 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.
• Limaçon with one cusp. The cusp is formed when the ratio of a to b in the equation is equal to one.
• An inverse curve of a parabola^[1] with focus as an inversion center^[2].
• An image of circle under complex map . ^[3]
• Sinusoidal spiral :

for

[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. Cardioid is one of the heart curves^[4]

[edit] Equations

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

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

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

is

.

See proof.

[edit] Examples

[edit] Mandelbrot set

There are many cardioids in the Mandelbrot set ^[5]:

• boundary of large central figure ( period 1 hyperbolic component) is a cardioid with equation :

• second largest cardioid is boundary of period 3 component on main antennae,

where

• generally 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

• Wittgenstein's rod
• microphone - for a discussion of cardioid microphones
• Loop antenna
• Radio direction finder
• Radio direction finding
• Yagi antenna

[edit] Bibliography

• Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. 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. ^ Weisstein, Eric W. "Heart Curve." From MathWorld--A Wolfram Web Resource.
5. ^ Boundary of hyperbolic components of Mandelbrot set

[edit] External links

Wikimedia Commons has media related to: Cardioids

• Hearty Munching on Cardioids at cut-the-knot
• Xah Lee, Cardioid (1998) (This site provides a number of alternative constructions).
• Jan Wassenaar, Cardioid, (2005)
• Cardioid at mathcurve.com