# Conchoid (mathematics)

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

expresses the given curve, then

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 , then the line's polar form is and therefore the conchoid can be expressed parametrically as

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

## 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

## External links

Media related to Conchoid at Wikimedia Commons