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cardioid generated by a rolling circle on a circle with the same radius

A cardioid (from the 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 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]

The name was coined by de Castillon in 1741[2] but had been the subject of study decades beforehand.[3] 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.

Contents

EquationsEdit

 
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

 
 

and the representation in

 .

Introducing the substitutions   and   one gets after removing the square root the implicit representation in

  .
Proof for the parametric representation

The proof can be done easily using complex numbers and their common description as 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   (origin) by an angle   can be performed by the multiplication of a point   (complex number) by  . Hence the

rotation   around point   is ,
rotation   around point   is:  .

A point   of the cardioid is generated by rotating the origin around point   and subsequent rotating around   by the same angle  :

 .

Herefrom one gets the parametric representation above:

 

(The following formulae   were used. See trigonometric functions.)

Metric propertiesEdit

For the cardioid as defined above the following formulas hold:

  • area  ,
  • arc length   and
  • radius of curvature  

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)

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 of a cardioidEdit

 
Chords of a cardioid

Chords through the cuspEdit

  • C1: chords through the cusp of the cardioid have the same length   .
  • C2: The midpoints of the chords through the cusp lie on the perimeter of the fixed generator circle (see picture) .
proof for 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

 
 ,

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 parabolaEdit

 
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 circlesEdit

 
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 parabolas'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

  •   (envelope condition).

(  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 linesEdit

 
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:  . (i.e.: The second point is moved by double velocity.)
  3. The envelope of these chords is a cardioid.
 
Cremona's generation of a cardioid
proof

The following consideration uses trigonometric formulae for  . In order to keep the calculations simple, the proof is given for the cardioid with polar representation   (see section 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:

  •  

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

Cardioid as caustic of a circleEdit

The considerations made in the previous section give a proof for the fact, 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   on the perimeter of a circle which is reflecting any ray, then the reflected rays within the circle are tangents of a cardioid.
proof

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 circleEdit

 
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

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 cardioidEdit

 
evolute of a cardioid
magenta: 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 (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 trajectoriesEdit

 
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:
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:

  and  

(The slope of any curve is dependent from   only, and not from the parameters   !)
Hence

 

That means: Any curve of the first pencil intersectcs any curve of the second pencil orthogonally.

 
4 cardioids in polar representation and their position in the coordinate system

Cardioids in different positionsEdit

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.

Cardioids in complex analysisEdit

 
Boundary of the central bulb of the Mandelbrot set is a 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 bulb 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.

 
The caustic appearing on the surface of this cup of coffee is a cardioid.

CausticsEdit

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.[4] 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 alsoEdit

ReferencesEdit

  1. ^ Weisstein, Eric W. "Parabola Inverse Curve". MathWorld.
  2. ^ Lockwood
  3. ^ Yates
  4. ^ "Surface Caustique" at Encyclopédie des Formes Mathématiques Remarquables

Further readingEdit

  • Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 24–25. ISBN 0-14-011813-6.

External linksEdit