# Conchoid (mathematics)

Conchoids of line with common center.
The fixed point O is the red dot, the black line is the given curve, and each pair of coloured curves is length d from the intersection with the line that a ray through O makes. In the blue case d is greater than O's distance from the line, so the upper blue curve loops back on itself. In the green case d is the same, and in the red case it's less.

A conchoid is a curve derived from a fixed point O, another curve, and a length d.

## 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 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 = α(θ) expresses the given curve then r = α(θ) ± d expresses the conchoid. For instance, if the curve is the straight line x = a, then r = acos(θ) and therefore the conchoid can be expressed as x = a ± d.cos(θ) and y = a.tan(θ) ± d.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.