In geometry, a cardioid (from Greek καρδιά (kardiá) '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 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.[1] 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.[2]

A cardioid
The caustic appearing on the surface of this cup of coffee is a cardioid.
Cardioid generated by a rolling circle on a circle with the same radius

The name was coined by Giovanni Salvemini in 1741[3] but the cardioid had been the subject of study decades beforehand.[4] Although 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 edit

Generation of a cardioid and the coordinate system used

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:
    and herefrom the representation in
  • polar coordinates:
  • Introducing the substitutions   and   one gets after removing the square root the implicit representation in Cartesian coordinates:

Proof for the parametric representation edit

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     and   were used.)

Metric properties edit

For the cardioid as defined above the following formulas hold:

  • area  ,
  • arc length   and
  • radius of curvature  

The proofs of these statements use in both cases the polar representation of the cardioid. For suitable formulas see polar coordinate system (arc length) and polar coordinate system (area)

Proof of the area formula

Proof of the arc length formula

Proof for the radius of curvature

The radius of curvature   of a curve in polar coordinates with equation   is (s. curvature)


For the cardioid   one gets


Properties edit

Chords of a cardioid

Chords through the cusp edit

Chords through the cusp of the cardioid have the same length  .
The midpoints of the chords through the cusp lie on the perimeter of the fixed generator circle (see picture).

Proof of C1 edit

The points   are on a chord through the cusp (=origin). Hence


Proof for C2 edit

For the proof the representation in the complex plane (see above) is used. For the points


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 edit

Cardioid generated by the inversion of a parabola across the unit circle (dashed)
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  . 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.

Cardioid as envelope of a pencil of circles edit

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:

  1. Choose a circle   and a point   on its perimeter,
  2. draw circles containing   with centers on  , and
  3. draw the envelope of these circles.
Proof with envelope condition

The envelope of the pencil of implicitly given curves

with parameter   consists of such points   which are solutions of the non-linear system
which is the envelope condition. Note that   means the partial derivative for parameter  .

Let   be the circle with midpoint   and radius  . Then   has parametric representation  . The pencil of circles with centers on   containing point   can be represented implicitly by

which is equivalent to
The second envelope condition is
One easily checks that the points of the cardioid with the parametric representation
fulfill the non-linear system above. The parameter   is identical to the angle parameter of the cardioid.

Cardioid as envelope of a pencil of lines edit

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:

  1. Draw a circle, divide its perimeter into equal spaced parts with   points (s. picture) and number them consecutively.
  2. Draw the chords:  . (That is, the second point is moved by double velocity.)
  3. The envelope of these chords is a cardioid.
Cremona's generation of a cardioid

Proof edit

The following consideration uses trigonometric formulae 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 r = 2(1 + cos 𝜑) edit

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 (1, 0) and radius 3 edit

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 edit

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.

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: light source  , light ray  , reflected ray  
Cardioid as caustic of a circle with light source (right) on the perimeter

Cardioid as caustic of a circle edit

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   on the perimeter of a circle which is reflecting any ray, then the reflected rays within the circle are tangents of a cardioid.

As in the previous section the circle may have midpoint   and radius  . Its parametric representation is

The tangent at circle point   has normal vector  . Hence the reflected ray has the normal vector   (see graph) and contains point  . The reflected ray is part of the line with equation (see previous section)
which is tangent of the cardioid with polar equation
from the previous section.

Remark: For such considerations usually multiple reflections at the circle are neglected.

Cardioid as pedal curve of a circle edit

Point of cardioid is foot of dropped perpendicular on tangent of circle

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 of a circle.

Proof edit

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 edit

  A cardioid
  Evolute of the cardioid
  One point P; its centre of curvature M; and its osculating circle.

The evolute 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, and facing the opposite direction (s. picture).

Proof edit

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 edit

Orthogonal cardioids

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 edit

For a curve given in polar coordinates 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:


(The slope of any curve depends on   only, and not on the parameters   or  !)


That means: Any curve of the first pencil intersects any curve of the second pencil orthogonally.
4 cardioids in polar representation and their position in the coordinate system

In different positions edit

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 edit

Boundary of the central, period 1, region of the Mandelbrot set is a precise cardioid.

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 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.

Cardioid formed by light on a watch dial.

Caustics edit

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.[5] The shape of the curve at the bottom of a cylindrical cup is half of a nephroid, which looks quite similar.

Generating a cardioid as pedal curve of a circle

See also edit

Notes edit

  1. ^ Weisstein, Eric W. "Parabola Inverse Curve". MathWorld.
  2. ^ S Balachandra Rao . Differential Calculus, p. 457
  3. ^ Lockwood
  4. ^ Yates
  5. ^ "Surface Caustique" at Encyclopédie des Formes Mathématiques Remarquables

References edit

  • R.C. Yates (1952). "Cardioid". A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 4 ff.
  • Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 24–25. ISBN 0-14-011813-6.

External links edit