# .

A conchoid is a curve derived from a fixed point O, another curve, and a length d.

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

$$r=\alpha(\theta)$$

expresses the given curve, then

$$r=\alpha(\theta)\pm d$$

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 $$r=\frac{a}{\cos \theta}$$ and therefore the conchoid can be expressed parametrically as

$$x=a \pm d \cos \theta,\, y=a \tan \theta \pm d \sin \theta.$$

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.

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.
"Conchoïde" at Encyclopédie des Formes Mathématiques Remarquables