Cardioid

Cartesian equation:
(x² + y² - 2ax)² = 4a²(x² + y²)
Polar equation:
r = 2a(1 + cos())

Click below to see one of the Associated curves.

Definitions of the Associated curves	Evolute
Involute 1	Involute 2
Inverse curve wrt origin	Inverse wrt another circle
Pedal curve wrt origin	Pedal wrt another point
Negative pedal curve wrt origin	Negative pedal wrt another point
Caustic wrt horizontal rays	Caustic curve wrt another point

Its length had been found by La Hire in 1708, and he therefore has some claim to be the discoverer of the curve. In the notation given above the length is 16a. It is a special case of the Limacon of Pascal (Etienne Pascal) and so, in a sense, its study goes back long before Castillon or La Hire.

There are exactly three parallel tangents 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 4a and the area of the cardioid is 6πa².

We can easily give parametric equations for the cardioid, namely

x = a(2cos(t) - cos(2t)), y = a(2sin(t) - sin(2t)).

The pedal curve of the cardioid, where the cusp point is the pedal point, is Cayley's Sextic.

If the cusp of the cardioid is taken as the centre of inversion, the cardioid inverts to a parabola.

The caustic of a cardioid, where the radiant point is taken to be the cusp, is a nephroid.

There are some other heart-shaped curves, sent to us by Kurt Eisemann (San
Diego State University, USA):

(i) The curve with Cartesian equation: y = 0.75 x^2/3 √(1 - x²). Show the picture

(ii) The curve with Polar equation: r = sin²(π/8 - /4). Show the picture

Other Web site:

Xah Lee

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/history/Curves/Cardioid.html