Conchoid (mathematics)

Conchoid of Nicomedes drawn by an apparatus illustrated in Eutocius' Commentaries on the works of Archimedes

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.^[1]

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 $r = a sec θ$ 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.