Conchoid (mathematics)

[[Image:Conchoid of Nicomedes.png|400px|right|thumb|Conchoids of line with common center. {{legend|red|Fixed point $O$}}

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

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In geometry, a conchoid is a curve derived from a fixed point $O$, another curve, and a length $1=d =$. It was invented by the ancient Greek mathematician Nicomedes.

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

The simplest expression uses polar coordinates with $d$ 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 $d >$, then the line's polar form is $d <$ 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.