(x2 + y2 - 2ax)2 = b2(x2 + y2)
r = b + 2a 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|
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.
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