In geometry, a conchoid is a curve derived from a fixed point O, another curve, and a length d. It was invented by the ancient Greek mathematician Nicomedes.
Description For every line through O that intersects the given curve at A the two points on the line which are d from A are on the conchoid. The conchoid is, therefore, the cissoid of the given curve and a circle of radius d and center O. They are called conchoids because the shape of their outer branches resembles conch shells. The simplest expression uses polar coordinates with O at the origin. If
r = α ( θ ) {\displaystyle r=\alpha (\theta )}
expresses the given curve, then
r = α ( θ ) ± d {\displaystyle r=\alpha (\theta )\pm d}
expresses the conchoid. If the curve is a line, then the conchoid is the conchoid of Nicomedes. For instance, if the curve is the line x = a, then the line's polar form is r = a sec θ and therefore the conchoid can be expressed parametrically as
x = a ± d cos θ , y = a tan θ ± d sin θ . {\displaystyle x=a\pm d\cos \theta ,\,y=a\tan \theta \pm d\sin \theta .}
A limaçon is a conchoid with a circle as the given curve. The so-called conchoid of de Sluze and conchoid of Dürer are not actually conchoids. The former is a strict cissoid and the latter a construction more general yet.
See also Cissoid Strophoid
References
J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 36, 49–51, 113, 137. ISBN 0-486-60288-5.
External links
conchoid with conic sections - interactive illustration Weisstein, Eric W. "Conchoid of Nicomedes". MathWorld. conchoid at mathcurves.com