![Herzkurve](https://upload.wikimedia.org/wikipedia/commons/a/a9/Herzkurve.svg)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a cardioid () is a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
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
In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an ''epicycle''—which rolls without slipping around a fixed circle. It is a particular kind of roulette.
Equati ...
having a single
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
. It is also a type of
sinusoidal spiral
In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates
:r^n = a^n \cos(n \theta)\,
where is a nonzero constant and is a rational number other than 0. With a rotation about the origin, ...
, and an
inverse curve
In inversive geometry, an inverse curve of a given curve is the result of applying an inverse operation to . Specifically, with respect to a fixed circle with center and radius the inverse of a point is the point for which lies on the ray ...
of the
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
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.
![Cardiod animation](https://upload.wikimedia.org/wikipedia/commons/d/d0/Cardiod_animation.gif)
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
An apple is an edible fruit produced by an apple tree (''Malus domestica''). Apple trees are cultivated worldwide and are the most widely grown species in the genus ''Malus''. The tree originated in Central Asia, where its wild ancestor, ' ...
without the stalk.
A
cardioid microphone
A microphone, colloquially called a mic or mike (), is a transducer that converts sound into an electrical signal. Microphones are used in many applications such as telephones, hearing aids, public address systems for concert halls and publi ...
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
![Kardioide](https://upload.wikimedia.org/wikipedia/commons/5/51/Kardioide.svg)
Let
be the common radius of the two generating circles with midpoints
,
the rolling angle and the origin the starting point (see picture). One gets the
*
parametric representation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric o ...
:
and herefrom the representation in
*
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
:
* Introducing the substitutions
and
one gets after removing the square root the implicit representation in
Cartesian coordinates:
Proof for the parametric representation
A proof can be established using complex numbers and their common description as the
complex plane. 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
(the origin) by an angle
can be performed by multiplying a point
(complex number) by
. Hence
: the rotation
around point
is
,
: the rotation
around point
is:
.
A point
of the cardioid is generated by rotating the origin around point
and subsequently rotating around
by the same angle
:
From here one gets the parametric representation above:
(The
trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
and
were used.)
Metric properties
For the cardioid as defined above the following formulas hold:
* ''area''
,
* ''arc length''
and
* ''
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
''
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
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
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
of the cardioid have the same length
.
; C2: The ''midpoints'' of the
chords
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 lie on the perimeter of the fixed generator circle (see picture).
Proof of C1
The points
are on a
chord through the cusp (=origin). Hence
Proof for C2
For the proof the representation in the complex plane (see above) is used. For the points
and
the midpoint of the chord
is
which lies on the perimeter of the circle with midpoint
and radius
(see picture).
Cardioid as inverse curve of a parabola
![Kardioide-parabel-1](https://upload.wikimedia.org/wikipedia/commons/0/0f/Kardioide-parabel-1.svg)
: A cardioid is the
inverse curve
In inversive geometry, an inverse curve of a given curve is the result of applying an inverse operation to . Specifically, with respect to a fixed circle with center and radius the inverse of a point is the point for which lies on the ray ...
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
. Hence the cardioid has the polar representation
and its inverse curve
which is a parabola (s.
parabola in polar coordinates) with the equation
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
In geometry, the cissoid of Diocles (; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be de ...
.
Cardioid as envelope of a pencil of circles
![Kardioide-kreise](https://upload.wikimedia.org/wikipedia/commons/a/a7/Kardioide-kreise.svg)
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
and a point
on its perimeter,
# draw circles containing
with centers on
, and
# draw the envelope of these circles.
Cardioid as envelope of a pencil of lines
![Kardioide-sehnen](https://upload.wikimedia.org/wikipedia/commons/9/97/Kardioide-sehnen.svg)
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
points (s. picture) and number them consecutively.
# Draw the chords:
. (That is, the second point is moved by double velocity.)
# The ''envelope'' of these chords is a cardioid.
![Cycloid-cremona-pr](https://upload.wikimedia.org/wikipedia/commons/6/6b/Cycloid-cremona-pr.svg)
Proof
The following consideration uses
trigonometric formulae
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
for
,
,
,
, and
.
In order to keep the calculations simple, the proof is given for the cardioid with polar representation
(''
§ Cardioids in different positions'').
= ''Equation of the tangent'' of the ''cardioid'' with polar representation
=
From the parametric representation
one gets the normal vector
. The equation of the tangent
is:
With help of trigonometric formulae and subsequent division by
, the equation of the tangent can be rewritten as:
= ''Equation of the chord'' of the ''circle'' with midpoint and radius
=
For the equation of the secant line passing the two points
one gets:
With help of trigonometric formulae and the subsequent division by
the equation of the secant line can be rewritten by:
= Conclusion
=
Despite the two angles
have different meanings (s. picture) one gets for
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:
is the pencil of secant lines of a circle (s. above) and
For fixed parameter t both the equations represent lines. Their intersection point is
which is a point of the cardioid with polar equation
Cardioid as caustic of a circle
The considerations made in the previous section give a proof that the
caustic
Caustic most commonly refers to:
* Causticity, a property of various corrosive substances
** Sodium hydroxide, sometimes called ''caustic soda''
** Potassium hydroxide, sometimes called ''caustic potash''
** Calcium oxide, sometimes called ''caus ...
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
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
![Kardioide-kreistangenten](https://upload.wikimedia.org/wikipedia/commons/7/7e/Kardioide-kreistangenten.svg)
The Cremona generation of a cardioid should not be confused with the following generation:
Let be
a circle and
a point on the perimeter of this circle. The following is true:
: The foots of perpendiculars from point
on the tangents of circle
are points of a cardioid.
Hence a cardioid is a special
pedal curve
A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to:
Computers and other equipment
* Footmouse, a foot-operated computer mouse
* In medical transcription, a pedal is used to control ...
of a circle.
Proof
In a Cartesian coordinate system circle
may have midpoint
and radius
. The tangent at circle point
has the equation
The foot of the perpendicular from point
on the tangent is point
with the still unknown distance
to the origin
. Inserting the point into the equation of the tangent yields
which is the polar equation of a cardioid.
''Remark:'' If point
is not on the perimeter of the circle
, one gets a
limaçon of Pascal.
The evolute of a cardioid
![Cardioid-evol](https://upload.wikimedia.org/wikipedia/commons/f/f7/Cardioid-evol.svg)
The
evolute
In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that cur ...
of a curve is the locus of centers of curvature. In detail: For a curve
with radius of curvature
the evolute has the representation
with
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
the unit normal is
and the radius of curvature
Hence the parametric equations of the evolute are
These equations describe a cardioid a third as large, rotated 180 degrees and shifted along the x-axis by
.
(Trigonometric formulae were used:
)
Orthogonal trajectories
![Cardioid-penc](https://upload.wikimedia.org/wikipedia/commons/2/25/Cardioid-penc.svg)
An
orthogonal trajectory 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
are the cardioids with equations
(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
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
by a function
the following connection to Cartesian coordinates hold:
and for the derivatives
Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point
:
For the cardioids with the equations
and
respectively one gets:
and
(The slope of any curve depends on
only, and not on the parameters
or
!)
Hence
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
![Mandel zoom 00 mandelbrot set](https://upload.wikimedia.org/wikipedia/commons/2/21/Mandel_zoom_00_mandelbrot_set.jpg)
In
complex analysis, the
image of any circle through the origin under the map
is a cardioid. One application of this result is that the boundary of the central period-1 component of the
Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
is a cardioid given by the
equation
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
In geometry, a nephroid () is a specific plane curve. It is a type of epicycloid in which the smaller circle's radius differs from the larger by a factor of one-half.
Name
Although the term ''nephroid'' was used to describe other curves, it was ...
, which looks quite similar.
See also
* Limaçon
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. I ...
* Nephroid
In geometry, a nephroid () is a specific plane curve. It is a type of epicycloid in which the smaller circle's radius differs from the larger by a factor of one-half.
Name
Although the term ''nephroid'' was used to describe other curves, it was ...
* Deltoid
* Wittgenstein's rod
Wittgenstein's rod is a problem in geometry discussed by 20th-century philosopher Ludwig Wittgenstein.
Description
A ray is drawn with its origin on a circle, through an external point and a point is chosen at some constant distance from t ...
* Cardioid microphone
A microphone, colloquially called a mic or mike (), is a transducer that converts sound into an electrical signal. Microphones are used in many applications such as telephones, hearing aids, public address systems for concert halls and publi ...
* Lemniscate of Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. ...
* Loop antenna
A loop antenna is a radio antenna consisting of a loop or coil of wire, tubing, or other electrical conductor, that is usually fed by a balanced source or feeding a balanced load. Within this physical description there are two (possibly three) d ...
* Radio direction finder
* Radio direction finding
* Yagi antenna Yagi may refer to:
Places
* Yagi, Kyoto, in Japan
* Yagi (Kashihara), in Nara Prefecture, Japan
* Yagi-nishiguchi Station, in Kashihara, Nara, Japan
* Kami-Yagi Station, a JR-West Kabe Line station located in 3-chōme, Yagi, Asaminami-ku, Hiroshima ...
* Giovanni Salvemini
Giovanni Francesco Mauro Melchiorre Salvemini di Castiglione FRS (15 January 1708 in Castiglione del Valdarno – 11 October 1791 in Berlin) was an Italian mathematician and astronomer.
Life
Salvemini was born on 15 January 1708 in Castiglion ...
Notes
References
*
*
External links
*
*
Hearty Munching on Cardioids
at cut-the-knot
*
*
*
* Xah Lee,
Cardioid
' (1998) ''(This site provides a number of alternative constructions)''.
* Jan Wassenaar,
', (2005)
{{Authority control
Roulettes (curve)