Cardioid

Cartesian equation: (x+y-2ax) = 4a(x+y)

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 (Étienne 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 6a.

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.

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