Fine Art


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 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



J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 36, 49–51, 113, 137. ISBN 0-486-60288-5.
"Conchoïde" at Encyclopédie des Formes Mathématiques Remarquables

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World