Cardioid

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.

[2] 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.

[5] 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).

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

The rolling movement of the black circle on the blue one can be split into two rotations.

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).

(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: The envelope of the pencil of implicitly given curves

One easily checks that the points of the cardioid with the parametric representation

A similar and simple method to draw a cardioid uses a pencil of lines.

It is due to L. Cremona: The following consideration uses trigonometric formulae for

In order to keep the calculations simple, the proof is given for the cardioid with polar representation

Hence any secant line of the circle, defined above, is a tangent of the cardioid, too: Remark: The proof can be performed with help of the envelope conditions (see previous section) of an implicit pencil of curves:

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.

The reflected ray is part of the line with equation (see previous section)

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

The following is true: Hence a cardioid is a special pedal curve of a circle.

These equations describe a cardioid a third as large, rotated 180 degrees and shifted along the x-axis by

For cardioids the following is true: (The second pencil can be considered as reflections at the y-axis of the first one.

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point

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, the image of any circle through the origin under the map

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.

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.

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