- Cardioid
Cardioid is

closed curve with onecusp .**Definition**In

geometry , the cardioid is anepicycloid with onecusp .**Construction***

epicycloid produced as the path (orlocus ) of a point on the circumference of acircle as that circle rolls around another fixed circle with the same radius.*

limaçon with one cusp. The cusp is formed when theratio of a to b in theequation is equal to one.*an

inverse curve of aparabola [*[*] with focus as an invesion center [*http://mathworld.wolfram.com/InverseCurve.html Weisstein, Eric W. "Inverse Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseCurve.html*]*[*] .*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 $partial\; D\; =\; left\{\; w:\; abs(2w)=1\; ight\; \}$ undercomplex 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 (orcusp ). It is rather shaped more like the outline of the cross section of aplum .**Equations**Since the cardioid is an

epicycloid with one cusp, incartesian coordinates it hasparametric equation s:$x(t)\; =\; 2r\; left(\; cos\; t\; -\; \{1\; over\; 2\}\; cos\; 2\; t\; ight)\; ,$

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

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 direction s, 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*] atcut-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)

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