Conchoid

Cartesian equation:
(x - b)2(x2 + y2) - a2x2 = 0
Polar equation:
r = a + b sec(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


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The name means shell form and was studied by the Greek mathematician Nicomedes in about 200 BC in relation to the problem of duplication of the cube. Nicomedes recognised the three distinct forms seen in this family.

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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Xah Lee


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JOC/EFR/BS January 1997

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