This quartic ^{1)}
curve has the form of a heart ^{2)}, so that
it is also called the heart curve ^{3)}. De
Castillon used the name cardioid for the first time, in a paper in the
Philosophical Transactions of the Royal Society (1741).
The curve is the epicycloid for which the rolling circle and the rolled circle
have the same radius. Besides, the curve can be seen as a special case of the limaçon, and
it is also a sinusoidal
spiral.
When light rays fall on a
concave mirror with a large angle, no focus point but a focus line is to be
seen. This
line is a cardioid. The same phenomenon can occur nearby a lamp, or as a result of light
falling in a cup of tea.
This seems to me the property of the cardioid that it is the catacaustic of the circle
(with the source on the circumference). The cardioid is also the pedal of the
circle (pedal point not on circle).
Other properties of the curve are:
The cardioid is the
envelope of the chords of a circle, between points P and Q, which follow the
circle in the same direction, where one point has the double speed of the other.
This construction is called the generation of Cremona.
This means in the figure that the points 10 and 20, 11 and 22, and so on, have
been connected.
Given a circle C
through the origin. Then the cardioid is the envelope of the circles with
as diameter the line through the origin and a point on C.
The first to study the curve was Römer (1674), followed by Vaumesle
(1678) and Koërsma (1689). And more extensively by Ozanam in
1691.
La Hire
found its length (4) in 1708.
Its equation can also be written as: r = cos^{2}φ/2.
The curve can be written in a Whewell equation as s = cos φ/3
^{3)}.
notes
1) equation: (x^{2} +y^{2}y)^{2} = x^{2} +y^{2}
2) kardia (Gr.) = heart
3) In German: Herzkurve
In Italian: cardioide.
