Cardioid

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

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


If your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves.

The cardioid, a name first used by de Castillon in a paper in the Philosophical Transactions of the Royal Societyin 1741, is a curve that is the locus of a point on the circumference of circle rolling round the circumference of a circle of equal radius. Of course the name means 'heart-shaped'.

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 x2/3 plusminus √(1 - x2). Show the picture

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

Other Web site:

Xah Lee


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JOC/EFR/BS January 1997

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