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 reflections of a fixed point on a circle through all tangents to the circle.
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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. It can also be defined as the set of points of reflections 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.
Contents
Equations
Let
a
{\displaystyle a}
be the common radius of the two generating circles with midpoints
(
−
a
,
0
)
,
(
a
,
0
)
{\displaystyle (-a,0),(a,0)}
,
φ
{\displaystyle \varphi }
the rolling angle and the origin the starting point (see picture). One gets the
parametric representation:
x
(
φ
)
=
2
a
(
1
−
cos
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
0
{\displaystyle 0}
(the origin) by an angle
φ
{\displaystyle \varphi }
can be performed by multiplying a point
z
{\displaystyle z}
(complex number) by
e
i
φ
{\displaystyle e^{i\varphi }}
. Hence
the rotation
Φ
+
{\displaystyle \Phi _{+}}
around point
a
{\displaystyle a}
is
:
z
↦
a
+
(
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
C1
Chords through the cusp of the cardioid have the same length
4
a
{\displaystyle 4a}
.
C2
The midpoints of the chords through the cusp lie on the perimeter of the fixed generator circle (see picture).
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
a
=
1
2
{\textstyle a={\frac {1}{2}}}
. Hence the cardioid has the polar representation
r
(
φ
)
=
1
−
cos
φ
{\displaystyle r(\varphi )=1-\cos \varphi }
and its inverse curve
r
(
φ
)
=
1
1
−
cos
φ
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:
Choose a circle
c
{\displaystyle c}
and a point
O
{\displaystyle O}
on its perimeter,
draw circles containing
O
{\displaystyle O}
with centers on
c
{\displaystyle c}
, and
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:
Draw a circle, divide its perimeter into equal spaced parts with
2
N
{\displaystyle 2N}
points (s. picture) and number them consecutively.
Draw the chords:
(
1
,
2
)
,
(
2
,
4
)
,
…
,
(
n
,
2
n
)
,
…
,
(
N
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.
If in the plane there is a light source at a point
Z
{\displaystyle Z}
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:
Let
k
{\displaystyle k}
be a circle and
O
{\displaystyle O}
a point on the perimeter of this circle. The following is true:
The foots of perpendiculars from point
O
{\displaystyle O}
on the tangents of circle
k
{\displaystyle k}
are points of a cardioid.
Hence a cardioid is a special pedal curve of a circle.
In a Cartesian coordinate system circle
k
{\displaystyle k}
may have midpoint
(
2
a
,
0
)
{\displaystyle (2a,0)}
The evolute of a cardioid
The evolute of a curve is the locus of centers of curvature. In detail: For a curve
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 of any circle through the origin under the map
z
→
z
2
{\displaystyle z\to z^{2}}
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.
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
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.
