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]

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

[math]\displaystyle{ r=\alpha(\theta) }[/math]

expresses the given curve, then

[math]\displaystyle{ r=\alpha(\theta)\pm d }[/math]

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

[math]\displaystyle{ x=a \pm d \cos \theta,\, y=a \tan \theta \pm d \sin \theta. }[/math]

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.

References

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

Conchoid (Mathematics)

Contents

Description

See also

References

External links