Conchoid (mathematics)

Short description: Curve traced by a line as it slides along another curve about a fixed point

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]

## Contents

* 1 Description
* 2 See also
* 3 References
* 4 External links

## 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)

expresses the given curve, then

r=α(θ)±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±dcos⁡θ,y=atan⁡θ±dsin⁡θ.

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

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

* J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 36, 49–51, 113, 137. ISBN 0-486-60288-5. https://archive.org/details/catalogofspecial00lawr/page/36.

## External links

* conchoid with conic sections \- interactive illustration
* Weisstein, Eric W.. "Conchoid of Nicomedes". http://mathworld.wolfram.com/ConchoidofNicomedes.html.
* conchoid at mathcurves.com

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