Cardioid

Cardioid is closed curve with one cusp.

Definition

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

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 [ [http://mathworld.wolfram.com/InverseCurve.html Weisstein, Eric W. "Inverse Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCurve.html ] ] with focus as an invesion center [ [http://mathworld.wolfram.com/ParabolaInverseCurve.html Weisstein, Eric W. "Parabola Inverse Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ParabolaInverseCurve.html ] ] .

* an image of circle $partial\; D\; =\; left\{\; w:\; abs(2w)=1\; ight\; \}$ under complex map $w\; o\; c\; =\; w-w^2\; ,$. [ [http://virtualmathmuseum.org/ConformalMaps/square2/index.html 3D-XplorMath Conformal Maps a*z^b+b*z ] ]
* Sinusoidal spiral : $r^n\; =\; a^n\; cos(n\; heta),$

::for$qquad\; n\; =\; frac\{1\}\{2\},$

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.

Equations

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

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

:$y(t)\; =\; 2r\; left(\; sin\; t\; -\; \{1\; over\; 2\}\; sin\; 2\; t\; ight)\; ,$

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

The polar equation

:$ho(t)\; =\; 2r(1\; -\; cos\; t).\; ,$

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

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 with polar equation:$ho\; (t)\; =\; a(1\; -\; cos\; t)\; ,$is:$A\; =\; \{3over\; 2\}\; pi\; a^2$.

"See proof."

Examples

Mandelbrot set

There are many cardioids in Mandelbrot set [] :
* boundary of large central figure ( period 1 hyperbolic component) is a cardioid with equation :

$c\; =\; frac\{e^\{it\{2\}\; -\; left\; (frac\{e^\{it\{2\}\; ight\; )^2\; ,$
* second largest cardioid is boundary of period 3 component on main antennae, $c\; =\; left\; (\; frac\{(P-1)sqrt\{27P^2-22P+23\{6sqrt\{3-frac\{27P^2-36P+25\}\{54\}\; ight\; )\; ^\{1/3\}+\; frac\{3P+1\}\{9left(frac\{\; (P-1)\; sqrt\{27P^2-22P+23\{6\; sqrt\{3\; -frac\{27P^2-36P+25\}\{54\}\; ight\; )^\{1/3\; -\; frac\{2\}\{3\}\; ,$

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

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

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.

ee also

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

Bibliography

*=References=

External links

* [http://www.cut-the-knot.org/ctk/Cardi.shtml Hearty Munching on Cardioids] at cut-the-knot
* Xah Lee, " [http://www.xahlee.org/SpecialPlaneCurves_dir/Cardioid_dir/cardioid.html Cardioid] " (1998) "(This site provides a number of alternative constructions)".
* Jan Wassenaar, " [http://www.2dcurves.com/roulette/rouletteca.html Cardioid] ", (2005)