The cardioid is a degenerate case of the limaçon. It is also a 1-cusped epicycloid (with ) and is the catacaustic
formed by rays originating at a point on the circumference of a circle
and reflected by the circle.
The name cardioid was first used by de Castillon in Philosophical Transactions of the Royal Society in 1741. Its arc length was
found by la Hire in 1708. There are exactly three paralleltangents to the cardioid with any given gradient. Also,
the tangents at the ends of any chord
through the cusp point are at right
angles. The length of any chord through the cusp
point is .
The cardioid may also be generated as follows. Draw a circle and fix a point on it. Now draw a set of circles
centered on the circumference of and passing through . The envelope of these circles
is then a cardioid (Pedoe 1995). Let the circle be centered at the origin and have radius
1, and let the fixed point be . Then the radius of a circle centered at an angle from (1, 0) is
(6)
(7)
(8)
If the fixed point
is not on the circle, then the resulting envelope is
a limaçon instead of a cardioid.