Conchoid (mathematics)

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]

Conchoids of line with common center.

  Fixed point O

  Given curve

Each pair of coloured curves is length d from the intersection with the line that a ray through O makes.

  d > distance of O from the line

  d = distance of O from the line

  d < distance of O from the line

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 x = a, then the line's polar form is r = a sec θ 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.

See also

• Cissoid
• Strophoid