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In geometry, a cardioid () 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 can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. A cardioid can also be defined as the set of points of reflections of a fixed point on a circle through all tangents to the circle. The name was coined by de Castillon in 1741 but the cardioid had been the subject of study decades beforehand.Yates Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk. A cardioid microphone exhibits an 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 around the microphone which is the "stalk" of the apple.


Equations

Let a be the common radius of the two generating circles with midpoints (-a,0), (a,0), \varphi the rolling angle and the origin the starting point (see picture). One gets the * parametric representation: \begin x(\varphi) &= 2a (1 - \cos\varphi)\cdot\cos\varphi \ , \\ y(\varphi) &= 2a (1 - \cos\varphi)\cdot\sin\varphi \ , \qquad 0\le \varphi < 2\pi \end and herefrom the representation in * polar coordinates: r(\varphi) = 2a (1 - \cos\varphi). * Introducing the substitutions \cos\varphi = x/r and r = \sqrt one gets after removing the square root the implicit representation in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
: \left(x^2 + y^2\right)^2 + 4 a x \left(x^2 + y^2\right) - 4a^2 y^2 = 0.


Proof for the parametric representation

A proof can be established using complex numbers and their common description as the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. The rolling movement of the black circle on the blue one can be split into two rotations. In the complex plane a rotation around point 0 (the origin) by an angle \varphi can be performed by multiplying a point z (complex number) by e^. Hence : the rotation \Phi_+ around point a is:z \mapsto a + (z - a)e^, : the rotation \Phi_- around point -a is: z \mapsto -a + (z + a)e^. A point p(\varphi) of the cardioid is generated by rotating the origin around point a and subsequently rotating around -a by the same angle \varphi: p(\varphi) = \Phi_ - (\Phi_+(0)) = \Phi_-\left(a - ae^\right) = -a + \left( a - ae^ + a\right)e^ = a\;\left(-e^ + 2e^ - 1\right). From here one gets the parametric representation above: \begin x(\varphi) &=& a\;(-\cos(2\varphi) + 2\cos\varphi - 1) &=& 2a(1 - \cos\varphi)\cdot\cos\varphi & & \\ y(\varphi) &=& a\;(-\sin(2\varphi) + 2\sin\varphi) &=& 2a(1 - \cos\varphi)\cdot\sin\varphi &.& \end (The trigonometric identities e^ = \cos\varphi + i\sin\varphi, \ (\cos\varphi)^2 + (\sin\varphi)^2 = 1, \cos(2\varphi) = (\cos\varphi)^2 - (\sin\varphi)^2, and \sin (2\varphi) = 2\sin\varphi\cos\varphi were used.)


Metric properties

For the cardioid as defined above the following formulas hold: * ''area'' A = 6\pi a^2, * ''arc length'' L = 16 a and * '' radius of curvature'' \rho(\varphi) = \tfraca\sin\tfrac \, . The proofs of these statement use in both cases the polar representation of the cardioid. For suitable formulas see polar coordinate system (arc length) and polar coordinate system (area)


Properties


Chords through the cusp

; C1: '' Chords'' through the cusp of the cardioid have the same length 4a. ; C2: The ''midpoints'' of the chords through the cusp lie on the perimeter of the fixed generator circle (see picture).


Proof of C1

The points P: p(\varphi),\; Q: p(\varphi + \pi) are on a
chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ( ...
through the cusp (=origin). Hence \begin , PQ, &= r(\varphi) + r(\varphi + \pi) \\ &= 2a (1 - \cos\varphi) + 2a (1 - \cos(\varphi + \pi)) = \cdots = 4a \end.


Proof for C2

For the proof the representation in the complex plane (see above) is used. For the points P:\ p(\varphi) = a\,\left(-e^ + 2e^ - 1\right) and Q:\ p(\varphi + \pi) = a\,\left(-e^ + 2e^ - 1\right) = a\,\left(-e^ - 2e^ - 1\right), the midpoint of the chord PQ is M:\ \tfrac(p(\varphi) + p(\varphi + \pi)) = \cdots = -a - ae^ which lies on the perimeter of the circle with midpoint -a and radius a (see picture).


Cardioid as inverse curve of a parabola

: A cardioid is the inverse curve of a parabola with its focus at the center of inversion (see graph) For the example shown in the graph the generator circles have radius a = \frac. Hence the cardioid has the polar representation r(\varphi) = 1 - \cos\varphi and its inverse curve r(\varphi) = \frac, which is a parabola (s. parabola in polar coordinates) with the equation x = \tfrac\left(y^2 - 1\right) in Cartesian coordinates. ''Remark: ''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.


Cardioid as envelope of a pencil of circles

In the previous section if one inverts additionally the tangents of the parabola one gets a pencil of circles through the center of inversion (origin). A detailed consideration shows: The midpoints of the circles lie on the perimeter of the fixed generator circle. (The generator circle is the inverse curve of the parabola's directrix.) This property gives rise to the following simple method to ''draw'' a cardioid: # Choose a circle c and a point O on its perimeter, # draw circles containing O with centers on c, and # draw the envelope of these circles.


Cardioid as envelope of a pencil of lines

A similar and simple method to draw a cardioid uses a pencil of ''lines''. It is due to L. Cremona: # Draw a circle, divide its perimeter into equal spaced parts with 2N points (s. picture) and number them consecutively. # Draw the chords: (1,2), (2,4), \dots, (n,2n), \dots, (N,2N), (N+1,2), (N+2,4), \dots . (That is, the second point is moved by double velocity.) # The ''envelope'' of these chords is a cardioid.


Proof

The following consideration uses trigonometric formulae for \cos\alpha + \cos\beta, \sin\alpha + \sin\beta, 1 + \cos 2\alpha , \cos 2\alpha, and \sin 2\alpha. In order to keep the calculations simple, the proof is given for the cardioid with polar representation r = 2(1 \mathbin \cos\varphi) ('' § Cardioids in different positions'').


= ''Equation of the tangent'' of the ''cardioid'' with polar representation r = 2(1 + \cos\varphi)

= From the parametric representation \begin x(\varphi) &= 2(1 + \cos\varphi) \cos \varphi, \\ y(\varphi) &= 2(1 + \cos\varphi) \sin \varphi \end one gets the normal vector \vec n = \left(\dot y , -\dot x\right)^\mathsf. The equation of the tangent \dot y(\varphi) \cdot (x - x(\varphi)) - \dot x(\varphi) \cdot (y - y(\varphi)) = 0 is: (\cos2\varphi + \cos \varphi)\cdot x + (\sin 2\varphi + \sin \varphi)\cdot y = 2(1 + \cos \varphi)^2 \, . With help of trigonometric formulae and subsequent division by \cos\frac\varphi, the equation of the tangent can be rewritten as: \cos(\tfrac\varphi) \cdot x + \sin\left(\tfrac\varphi\right) \cdot y = 4 \left(\cos\tfrac\varphi\right)^3 \quad 0 < \varphi < 2\pi,\ \varphi \ne \pi .


= ''Equation of the chord'' of the ''circle'' with midpoint (1,0) and radius 3

= For the equation of the secant line passing the two points (1 + 3\cos\theta, 3\sin\theta),\ (1 + 3\cos\theta, 3\sin\theta)) one gets: (\sin\theta - \sin 2\theta) x + (\cos 2\theta - \sin \theta) y = -2\cos \theta - \sin(2\theta) \, . With help of trigonometric formulae and the subsequent division by \sin\frac\theta the equation of the secant line can be rewritten by: \cos\left(\tfrac\theta\right) \cdot x + \sin\left(\tfrac\theta\right) \cdot y = 4 \left(\cos\tfrac\theta\right)^3 \quad 0 < \theta < 2\pi .


= Conclusion

= Despite the two angles \varphi, \theta have different meanings (s. picture) one gets for \varphi = \theta the same line. Hence any secant line of the circle, defined above, is a tangent of the cardioid, too: : ''The cardioid is the envelope of the chords of a circle.'' ''Remark:''
The proof can be performed with help of the ''envelope conditions'' (see previous section) of an implicit pencil of curves: F(x, y, t) = \cos\left(\tfract\right) x + \sin\left(\tfract\right) y - 4 \left(\cos\tfract\right)^3 = 0 is the pencil of secant lines of a circle (s. above) and F_t(x, y, t) = - \tfrac\sin\left(\tfract\right) x + \tfrac\cos \left(\tfract\right) y + 3\cos\left(\tfract\right) \sin t = 0\, . For fixed parameter t both the equations represent lines. Their intersection point is x(t) = 2(1 + \cos t)\cos t,\quad y(t) = 2(1 + \cos t)\sin t, which is a point of the cardioid with polar equation r = 2(1 + \cos t).


Cardioid as caustic of a circle

The considerations made in the previous section give a proof that the caustic of a circle with light source on the perimeter of the circle is a cardioid. : If in the plane there is a light source at a point Z on the perimeter of a circle which is reflecting any ray, then the reflected rays within the circle are tangents of a cardioid. ''Remark:'' For such considerations usually multiple reflections at the circle are neglected.


Cardioid as pedal curve of a circle

The Cremona generation of a cardioid should not be confused with the following generation: Let be k a circle and O a point on the perimeter of this circle. The following is true: : The foots of perpendiculars from point O on the tangents of circle k are points of a cardioid. Hence a cardioid is a special pedal curve of a circle.


Proof

In a Cartesian coordinate system circle k may have midpoint (2a,0) and radius 2a. The tangent at circle point (2a + 2a\cos\varphi, 2a\sin \varphi) has the equation (x - 2a) \cdot \cos\varphi + y\cdot\sin\varphi = 2a\, . The foot of the perpendicular from point O on the tangent is point (r\cos \varphi, r\sin \varphi) with the still unknown distance r to the origin O. Inserting the point into the equation of the tangent yields (r\cos\varphi - 2a)\cos\varphi + r\sin^2\varphi = 2a \quad \rightarrow \quad r = 2a(1 + \cos \varphi) which is the polar equation of a cardioid. ''Remark:'' If point O is not on the perimeter of the circle k, one gets a
limaçon of Pascal In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. ...
.


The evolute of a cardioid

The evolute of a curve is the locus of centers of curvature. In detail: For a curve \vec x(s) = \vec c(s) with radius of curvature \rho(s) the evolute has the representation \vec X(s) = \vec c(s) + \rho(s)\vec n(s). with \vec n(s) the suitably oriented unit normal. For a cardioid one gets: : The ''evolute'' of a cardioid is another cardioid one third as large (s. picture).


Proof

For the cardioid with parametric representation x(\varphi) = 2a (1 - \cos\varphi)\cos\varphi = 4a \sin^2\tfrac\cos\varphi\, , y(\varphi) = 2a (1 - \cos\varphi)\sin\varphi = 4a \sin^2\tfrac\sin\varphi the unit normal is \vec n(\varphi) = (-\sin\tfrac\varphi, \cos\tfrac\varphi) and the radius of curvature \rho(\varphi) = \tfraca\sin\tfrac \, . Hence the parametric equations of the evolute are X(\varphi) = 4a \sin^2\tfrac\cos\varphi-\tfraca\sin\tfrac\cdot \sin\tfrac \varphi = \cdots = \tfraca\cos^2\tfrac\cos\varphi - \tfraca \, , Y(\varphi) = 4a \sin^2\tfrac \sin\varphi + \tfraca \sin\tfrac \cdot\cos\tfrac \varphi = \cdots = \tfraca \cos^2\tfrac \sin\varphi \, . These equations describe a cardioid a third as large, rotated 180 degrees and shifted along the x-axis by -\tfrac a. (Trigonometric formulae were used: \sin\tfrac\varphi = \sin\tfrac\cos\varphi + \cos\tfrac\sin\varphi\ ,\ \cos\tfrac\varphi = \cdots, \ \sin\varphi = 2\sin\tfrac\cos\tfrac, \ \cos\varphi= \cdots \ . )


Orthogonal trajectories

An
orthogonal trajectory In mathematics an orthogonal trajectory is a curve, which intersects any curve of a given pencil of (planar) curves ''orthogonally''. For example, the orthogonal trajectories of a pencil of ''concentric circles'' are the lines through their commo ...
of a pencil of curves is a curve which intersects any curve of the pencil orthogonally. For cardioids the following is true: : The orthogonal trajectories of the pencil of cardioids with equations r=2a(1-\cos\varphi)\ , \; a>0 \ , \ are the cardioids with equations r=2b(1+\cos\varphi)\ , \; b>0 \ . (The second pencil can be considered as reflections at the y-axis of the first one. See diagram.)


Proof

For a curve given in polar coordinates by a function r(\varphi) the following connection to Cartesian coordinates hold: \begin x(\varphi) &= r(\varphi)\cos\varphi\, ,\\ y(\varphi) &= r(\varphi)\sin\varphi \end and for the derivatives \begin \frac &= r'(\varphi)\cos\varphi - r(\varphi)\sin\varphi\, ,\\ \frac &= r'(\varphi)\sin\varphi + r(\varphi)\cos\varphi\, . \end Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r(\varphi), \varphi): \frac = \frac. For the cardioids with the equations r=2a(1-\cos\varphi) \; and r = 2b(1 + \cos\varphi)\ respectively one gets: \frac = \frac and \frac = -\frac\ . (The slope of any curve depends on \varphi only, and not on the parameters a or b!) Hence \frac\cdot \frac = \cdots = -\frac = -\frac = -1\, . That means: Any curve of the first pencil intersectcs any curve of the second pencil orthogonally.


In different positions

Choosing other positions of the cardioid within the coordinate system results in different equations. The picture shows the 4 most common positions of a cardioid and their polar equations.


In complex analysis

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of any circle through the origin under the map z \to z^2 is a cardioid. One application of this result is that the boundary of the central period-1 component of the Mandelbrot set is a cardioid given by the
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
c \,=\, \frac. 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.


Caustics

Certain 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."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.


See also

* Limaçon * Nephroid *
Deltoid Deltoid (delta-shaped) can refer to: * The deltoid muscle, a muscle in the shoulder * Kite (geometry), also known as a deltoid, a type of quadrilateral * A deltoid curve, a three-cusped hypocycloid * A leaf shape * The deltoid tuberosity, a part of ...
* Wittgenstein's rod * Cardioid microphone * Lemniscate of Bernoulli * Loop antenna *
Radio direction finder Direction finding (DF), or radio direction finding (RDF), isin accordance with International Telecommunication Union (ITU)defined as radio location that uses the reception of radio waves to determine the direction in which a radio station ...
*
Radio direction finding Direction finding (DF), or radio direction finding (RDF), isin accordance with International Telecommunication Union (ITU)defined as radio location that uses the reception of radio waves to determine the direction in which a radio station ...
* Yagi antenna * Giovanni Salvemini


Notes


References

* *


External links

* *
Hearty Munching on Cardioids
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
* * * * Xah Lee,
Cardioid
' (1998) ''(This site provides a number of alternative constructions)''. * Jan Wassenaar,

', (2005) {{Authority control Roulettes (curve)