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 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

$r=\alpha(\theta)$

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 $x=a$, then the line's polar form is $r=\frac{a}{\cos \theta}$ 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

References

• "Conchoïde" at Encyclopédie des Formes Mathématiques Remarquables

External links

Media related to at Wikimedia Commons