Limacon of Pascal

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

Polar equation: r = b + 2a 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

The name 'limacon' comes from the Latin limax meaning 'a snail'. Étienne Pascal corresponded with Mersenne whose house was a meeting place for famous geometers including Roberval.

Dürer should really be given the credit for discovering the curve since he gave a method for drawing the limacon, although he did not call it a limacon, in Underweysung der Messung published in 1525.

When b = 2a then the limacon becomes a cardioid while if b = a then it becomes a trisectrix. Notice that this trisectrix is not the Trisectrix of Maclaurin.

If b 2a then the area of the limacon is (2a+k). If b = a (the case drawn above with a = b = 1) then the area of the inner loop is a(-33/2) and the area between the loops is a(+33).

The limacon is an anallagmatic curve.

The limacon is also the catacaustic of a circle when the light rays come from a point a finite (non-zero) distance from the circumference. This was shown by Thomas de St Laurent in 1826.

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