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Wikipedia / Cardioid

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. It can also be defined as the set of points of reflectionss of a fixed point on a circle through all tangents to the circle.

Giovanni Salvemini coined the name cardioid in 1741, but the cardioid had been the subject of study decades beforehand. Although named for its resemblance to a conventional 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

the rolling angle and the origin the starting point (see picture). One gets the

Proof for the parametric representation

A proof can be established using complex numbers and their common description as the complex plane (complex number. Hence

  1. ,

  2. .

(The trigonometric identities were used.)

Metric properties

For the cardioid as defined above the following formulas hold:

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)

Properties

Chords through the cusp

  1. C1: Chords through the cusp.

  2. C2: The midpoints of the chords through the cusp lie on the perimeter of the fixed generator circle (see picture).

Proof of C1

are on a chord

Proof for C2

(see picture).

Cardioid as inverse curve of a parabola

  1. A cardioid is the inverse curve of a parabola with its focus at the center of inversion (see graph)

which is a parabola (s. parabola in polar coordinates 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:

  1. on its perimeter,

  2. , and

  3. 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:

  1. points (s. picture) and number them consecutively.

  2. . (That is, the second point is moved by double velocity.)

  3. The envelope of these chords is a cardioid.

Proof

The following consideration uses trigonometric formulae (§ Cardioids in different positions).

Equation of the tangent of the cardioid with polar representation r 2(1 + cos)
Equation of the chord of the circle with midpoint (1,0) and radius 3

For the equation of the secant line

Conclusion

the same line. Hence any secant line of the circle, defined above, is a tangent of the cardioid, too:

  1. The cardioid is the envelope of the chords of a circle.

Remark: The proof can be performed with help of the envelope conditions

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.

  1. 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:

a point on the perimeter of this circle. The following is true:

  1. are points of a cardioid.

Hence a cardioid is a special pedal curve of a circle.

Proof

then gives us our desired cardioid. Remark:, one gets a limaçon of Pascal.

The evolute of a cardioid

The evolute the suitably oriented unit normal.

For a cardioid one gets:

  1. The evolute of a cardioid is another cardioid, one third as large, and facing the opposite direction (s. picture).

Proof

.

)

Orthogonal trajectories

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 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

!)

That means: Any curve of the first pencil intersects 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, the image 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.

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

See also

Notes

References