Conchoid (mathematics)

File:Conchoid of Nicomedes.png

Conchoids of line with common center.

  Fixed point Template:Mvar

  Given curve

Each pair of coloured curves is length Template:Mvar from the intersection with the line that a ray through Template:Mvar makes.

  Template:Math distance of Template:Mvar from the line

  Template:Mvar distance of Template:Mvar from the line

  Template:Math distance of Template:Mvar from the line

File:Nicomedes.gif

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 Template:Mvar, another curve, and a length Template:Mvar. It was invented by the ancient Greek mathematician Nicomedes.

Description

For every line through Template:Mvar that intersects the given curve at Template:Mvar the two points on the line which are Template:Mvar from Template:Mvar are on the conchoid. The conchoid is, therefore, the cissoid of the given curve and a circle of radius Template:Mvar and center Template:Mvar. They are called conchoids because the shape of their outer branches resembles conch shells.

The simplest expression uses polar coordinates with Template:Mvar 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 Template:Math, then the line's polar form is Template:Math 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.

See also

• Cissoid
• Strophoid

External links

• conchoid with conic sections - interactive illustration
• conchoid at mathcurves.com