Conchoid

Cartesian equation: (x-b)(x+y)-ax = 0

Polar equation: r = a + b sec()

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

Nicomedes was a minor geometer who worked around 180 BC. His main invention was the conchoid ascribed to him by Pappus. It was a favourite with 17 C mathematicians and could be used, as Nicomedes had intended, to solve the problems of duplicating the cube and trisecting an angle.

Newton said it should be a 'standard' curve.

The conchoid has x = b as an asymptote and the area between either branch and the asymptote is infinite. The area of the loop is
b(a-b) - 2ab log((a+a-b)/b) + acos(b/a).

The conchoid was used in the construction of ancient buildings. The vertical section of columns was made in the shape of the loop of the conchoid.

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