(x - b)2(x2 + y2) - a2x2 = 0
r = a + b sec(θ)
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|Definitions of the Associated curves||Evolute|
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|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 Century 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√(a2 - b2) - 2ab log((a + √a2 - b2)/b) + a2cos-1(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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