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Conchoid (mathematics)

## Summary

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.[1]

Conchoids of line with common center.
The fixed point O is the red dot, the black line is the given curve, and each pair of coloured curves is length d from the intersection with the line that a ray through O makes. In the blue case d is greater than O's distance from the line, so the upper blue curve loops back on itself. In the green case d is the same, and in the red case it's less.
Conchoid of Nicomedes drawn by an apparatus illustrated in Eutocius' Commentaries on the works of Archimedes

## 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

${\displaystyle r=\alpha (\theta )}$

expresses the given curve, then

${\displaystyle r=\alpha (\theta )\pm d}$

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 ${\displaystyle x=a}$ , then the line's polar form is ${\displaystyle r=a\sec \theta }$  and therefore the conchoid can be expressed parametrically as

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

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.