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 $partial D = left\left\{ w: abs\left(2w\right)=1 ight \right\}$ 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 : $r^n = a^n cos\left(n heta\right),$

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

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

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

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

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

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

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:$ho \left(t\right) = a\left(1 - cos t\right) ,$is:$A = \left\{3over 2\right\} pi a^2$.

"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 :

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

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

* 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=

* [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)

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Look at other dictionaries:

• Cardioid — Car di*oid, n. [Gr. kardio eidh s heart shaped; kardi a heart + e i^dos shape.] (Math.) An algebraic curve, so called from its resemblance to a heart. [1913 Webster] …   The Collaborative International Dictionary of English

• cardioid — [kär′dē oid΄] n. [Gr kardioeidēs, heart shaped < kardia, HEART + oeidēs, OID] Math. a curve more or less in the shape of a heart, traced by a point on the circumference of a circle that rolls around the circumference of another equal circle …   English World dictionary

• cardioid — noun 1》 Mathematics a heart shaped curve traced by a point on the circumference of a circle as it rolls around another identical circle. 2》 a directional microphone with a pattern of sensitivity of this shape. adjective of the shape of a cardioid …   English new terms dictionary

• cardioid — noun Date: 1753 a heart shaped curve that is traced by a point on the circumference of a circle rolling completely around an equal fixed circle and has an equation in one of the forms ρ = a(1 ± cos θ) or ρ = a(1 ± sin θ) in polar coordinates …   New Collegiate Dictionary

• cardioid — heart shaped …   Dictionary of ichthyology

• cardioid — /kahr dee oyd /, n. Math. a somewhat heart shaped curve, being the path of a point on a circle that rolls externally, without slipping, on another equal circle. Equation: r = a (1 cosA). [1745 55; < Gk kardioeidés heart shaped. See CARDI , OID] * …   Universalium

• cardioid — 1. noun An epicycloid with exactly one cusp; the plane curve with polar equation having a shape supposedly heart shaped 2. adjective Having this characteristic shape …   Wiktionary

• cardioid — Resembling a heart. [cardi + G. eidos, resemblance] * * * car·di·oid (kahrґde oid) heartlike; resembling a heart …   Medical dictionary

• cardioid — heart shaped Shapes and Resemblance …   Phrontistery dictionary

• cardioid — n. heart shaped geometrical figure (Mathematics) …   English contemporary dictionary