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| [[Image:Cardioid animation.gif|right|thumb|300px|A cardioid generated by a rolling circle around another circle and tracing one point on the edge of it.]]
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| [[Image:CardioidCircleEnvelope.svg|300px|thumb|right|A cardioid given as the envelope of circles whose centers lie on a given circle and which pass through a fixed point on the given circle.]]
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| A '''cardioid''' (from the [[Greek language|Greek]] καρδία "heart") is a [[plane curve]] traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is therefore a type of [[limaçon]] and can also be defined as an [[epicycloid]] having a single [[Cusp (singularity)|cusp]]. It is also a type of [[sinusoidal spiral]], and an [[inverse curve]] of the [[parabola]] with the focus as the center of inversion.<ref>{{MathWorld|title=Parabola Inverse Curve|urlname=ParabolaInverseCurve}}</ref>
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| The name was coined by [[Giovanni Salvemini|de Castillon]] in 1741<ref>Lockwood</ref> but had been the subject of study decades beforehand.<ref name="Yates">Yates</ref> Named for its heart-like form, it is shaped more like the outline of the cross section of a round [[apple]] without the stalk.
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| A [[cardioid microphone]] exhibits an [[Acoustics|acoustic]] pickup pattern that, when graphed in two dimensions, resembles a cardioid, (any 2d plane containing the 3d straight line of the microphone body.) In three dimensions, the cardioid is shaped like an apple centred on the microphone which is the "stalk" of the apple.
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| ==Equations==
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| Based on the rolling circle description, with the fixed circle having the origin as its center, and both circles having radius ''a'', the cardioid is given by the following [[parametric equation]]s:
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| :<math> x = a (2\cos t - \cos 2 t), \,</math>
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| :<math> y = a (2\sin t - \sin 2 t). \,</math>
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| In the [[complex plane]] this becomes
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| :<math> z = a (2e^{it} - e^{2it}). \,</math>
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| Here ''a'' is the radius of the circles which generate the curve, and the fixed circle is centered at the origin. The point generating the curve touches the fixed circle at (''a'', 0), the cusp. The parameter ''t'' can be eliminated giving
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| :<math>(z\bar{z}-a^2)^2 -4a^2(z-a)(\bar{z}-a)=0</math>
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| or, in rectangular coordinates,
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| :<math>(x^2+y^2-a^2)^2-4a^2((x-a)^2+y^2)=0.\,</math>
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| {{-}}
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| {{Sinusoidal spirals.svg}}
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| These equations can be simplified somewhat by shifting the fixed circle to the right ''a'' units and choosing the point on the rolling circle so that it touches the fixed circle at the origin; this changes the orientation of the curve so that the cusp is on the left. The parametric equations are then:
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| :<math> x = a (1 + 2\cos t + \cos 2 t), \,</math>
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| :<math> y = a (2\sin t + \sin 2 t), \,</math>
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| or, in the complex plane,
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| :<math> z = a (1 + 2e^{it} + e^{2it}) = a(1 + e^{it})^2. \,</math>
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| With the substitution ''u''=tan ''t''/2,
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| :<math> e^{it} = \frac{1+iu}{1-iu},</math>
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| giving a rational parameterization:
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| :<math> z = \frac{4a}{(1-iu)^2}, </math>
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| or
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| :<math> x = \frac{4a(1-u^2)}{(1+u^2)^2}, </math>
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| :<math> y = \frac{8au}{(1+u^2)^2}.</math>
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| The parametrization can also be written
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| :<math> z = e^{it} 2a (1+\cos t), \,</math>
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| and in this form it is apparent that the equation for this cardioid may be written in [[polar coordinates]] as | |
| :<math> r = 2a(1 + \cos \theta)\,</math>
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| where θ replaces the parameter ''t''.
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| This can also be written
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| :<math> r = 4a\cos^2 \frac{\theta}{2}\,</math>
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| which implies that the curve is a member of the family of [[sinusoidal spiral]]s.
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| In [[Cartesian coordinates]], the equation for this cardioid is
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| :<math> \left(x^2+y^2-2ax\right)^2 = 4a^2\left(x^2 + y^2\right).\,</math>
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| ==Metrical properties==
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| The area enclosed by a cardioid can be computed from the polar equation:
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| :<math> A = 6 \pi a^2.</math>
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| This is 6 times the area of the circles used in the construction with rolling circles,<ref name="Yates">Yates</ref> or 1.5 times the area of the circle used in the construction with circle and tangent lines.
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| The [[arc length]] of a cardioid can be computed exactly,a rarity for algebraic curves. The total length is<ref name="Yates"/>
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| :<math> L = 16 a.</math>
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| ==Inverse curve==
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| [[Image:Inverse Curves Parabola Cardioid.svg|right|thumb|200px|The green cardioid is obtained by [[Inversive geometry|inverting]] the red [[parabola]] across the dashed [[circle]].]]
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| The cardioid is one possible [[inverse curve]] for a [[parabola]]. Specifically, if a parabola is [[inversive geometry|inverted]] across any [[circle]] whose center lies at the focus of the parabola, the result is a cardioid. The cusp of the resulting cardioid will lie at the center of the circle, and corresponds to the [[vanishing point]] of the parabola.
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| In terms of [[stereographic projection]], this says that a parabola in the [[Euclidean plane]] is the projection of a cardioid drawn on the [[sphere]] whose cusp is at the north pole.
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| Not every inverse curve of a parabola is a cardioid. For example, if a parabola is inverted across a circle whose center lies at the vertex of the parabola, then the result is a [[cissoid of Diocles]].
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| The picture to the right shows the parabola with polar equation
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| :<math>\rho(\theta) \,=\, \frac{1}{1 - \cos \theta}.\,</math>
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| In [[Cartesian coordinates]], this is the parabola <math>y^2 = 2x+1</math>. When this parabola is inverted across the [[unit circle]], the result is a cardioid with the [[Multiplicative inverse|reciprocal]] equation
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| :<math>\rho(\theta) \,=\, 1 - \cos \theta.\,</math>
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| [[Image:Mandel zoom 00 mandelbrot set.jpg|thumb|right|200px|[[Boundary (topology)|Boundary]] of the central bulb of the [[Mandelbrot set]] is a cardioid.]]
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| ==Cardioids in complex analysis==
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| In [[complex analysis]], the [[image (mathematics)|image]] of any circle through the origin under the map <math>z\to z^2</math> is a cardioid. One application of this result is that the boundary of the central bulb of the [[Mandelbrot set]] is a cardioid given by the [[Parametric equation|equation]]
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| :<math> c \,=\, \frac{1 - \left(e^{it}-1\right)^2}{4}.\,</math>
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| The Mandelbrot set contains an infinite number of slightly distorted copies of itself and the central bulb of any of these smaller copies is an approximate cardioid.
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| [[Image:Caustique.jpg|thumb|right|200px|The [[caustic (optics)|caustic]] appearing on the surface of this cup of coffee is a cardioid.]]
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| ==Caustics==
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| Certain [[Caustic (mathematics)|caustics]] can take the shape of cardioids. The catacaustic of a circle with respect to a point on the circumference is a cardioid. Also, the catacaustic of a cone with respect to rays parallel to a generating line is a surface whose cross section is a cardioid. This can be seen, as in the photograph to the right, in a conical cup partially filled with liquid when a light is shining from a distance and at an angle equal to the angle of the cone.<ref>[http://www.mathcurve.com/surfaces/caustic/caustic.shtml "Surface Caustique" at Encyclopédie des Formes Mathématiques Remarquables]</ref> The shape of the curve at the bottom of a cylindrical cup is half of a [[nephroid]], which looks quite similar.
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| [[Image:Cardioid construction.gif|right|thumb|210px|Generating a cardioid using a circle and tangent lines]]
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| ==See also== | |
| * [[Nephroid]]
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| * [[Deltoid curve|Deltoid]]
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| * [[Wittgenstein's rod]]
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| * [[Cardioid microphone]]
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| * [[Lemniscate of Bernoulli]]
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| * [[Loop antenna]]
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| * [[Radio direction finder]]
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| * [[Radio direction finding]]
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| * [[Yagi antenna]]
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| * [[Giovanni Salvemini]]
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| * [[Bonne projection|Bonne's Projection]]
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| ==References==
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| {{reflist}}
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| * {{MathWorld|title=Cardioid|urlname=Cardioid}}
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| * {{cite book | author=R.C. Yates | title=A Handbook on Curves and Their Properties | location=Ann Arbor, MI | publisher=J. W. Edwards | pages=4 ff.|chapter=Cardioid|year=1952 }}
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| ==Further reading==
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| * {{cite book | author = Wells D | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = 24–25}}
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| ==External links==
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| {{commons category|Cardioids}}
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| * {{MacTutor|class=Curves|id=Cardioid|title=Cardioid}}
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| * [http://www.cut-the-knot.org/ctk/Cardi.shtml Hearty Munching on Cardioids] at [[cut-the-knot]]
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| * {{MathWorld|title=Epicycloid--1-Cusped|urlname=Epicycloid1-Cusped}}
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| * {{MathWorld|title=Heart Curve|urlname=HeartCurve}}
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| * [http://www.mathcurve.com/courbes2d/cardioid/cardioid.shtml "Cardioid"] at Encyclopédie des Formes Mathématiques Remarquables
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| * Xah Lee, ''[http://www.xahlee.org/SpecialPlaneCurves_dir/Cardioid_dir/cardioid.html Cardioid]'' (1998) ''(This site provides a number of alternative constructions)''.
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| * Jan Wassenaar, ''[http://www.2dcurves.com/roulette/rouletteca.html Cardioid]'', (2005)
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| [[Category:Curves]]
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