Limacon of Pascal

Cartesian equation:
(x² + y² - 2ax)² = b²(x² + y²)
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 Messungpublished 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 (2a² + k²)π. If b = a (the case drawn above with a = b = 1) then the area of the inner loop is a²(π - 3√3/2) and the area between the loops is a²(π + 3√3).

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.
Other Web site:

Xah Lee

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