Conchoid (mathematics)

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Conchoids of line with common center.
  Fixed point O
  Given curve
Each pair of coloured curves is length d from the intersection with the line that a ray through O makes.
  d > distance of O from the line
  d = distance of O from the line
  d < distance of O from the line
Conchoid of Nicomedes drawn by an apparatus illustrated in Eutocius' Commentaries on the works of Archimedes

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

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 = α ( θ ) {\displaystyle r=\alpha (\theta )} {\displaystyle r=\alpha (\theta )}

expresses the given curve, then

r = α ( θ ) ± d {\displaystyle r=\alpha (\theta )\pm d} {\displaystyle 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 = a sec θ and therefore the conchoid can be expressed parametrically as

x = a ± d cos ⁡ θ , y = a tan ⁡ θ ± d sin ⁡ θ . {\displaystyle x=a\pm d\cos \theta ,\,y=a\tan \theta \pm d\sin \theta .} {\displaystyle 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

References

  1. Chisholm, Hugh, ed. (1911). "Conchoid" . Encyclopædia Britannica. Vol. 6 (11th ed.). Cambridge University Press. pp. 826–827.