Conchoid (mathematics)

A conchoid is a curve derived from a fixed point O, another curve, and a length d. 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 a circle with center O and the given curve. 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 = α(θ) ± d expresses the conchoid. Parametrically, it can be expressed as x = a + cos(θ) and y = atan(θ) + sin(θ).

All conchoids are cissoids with a circle centered on O as one of the curves.

The prototype of this class is the conchoid of Nicomedes in which the given curve is a line.

A limaçon is a conchoid with a circle as the given curve.

The often-so-called conchoid of de Sluze and conchoid of Dürer do not fit this definition; the former is a strict cissoid and the latter a construction more general yet.

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