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Conchoid (mathematics)

{{short description|Curve traced by a line as it slides along another curve about a fixed point}}

[[Image:Conchoid of Nicomedes.png|400px|right|thumb|Conchoids of line with common center.

{{legend|red|Fixed point {{mvar|O}}}}

{{legend-line|solid #333333|Given curve}}

Each pair of coloured curves is length {{mvar|d}} from the intersection with the line that a ray through {{mvar|O}} makes.

{{legend-line|solid blue|{{math|d > }} distance of {{mvar|O}} from the line}}

{{legend-line|solid lime|{{mvar|1=d = }} distance of {{mvar|O}} from the line}}

{{legend-line|solid red|{{math|d < }} distance of {{mvar|O}} from the line}}]]

File:Nicomedes.gif

In geometry, a conchoid is a curve derived from a fixed point {{mvar|O}}, another curve, and a length {{mvar|d}}. It was invented by the ancient Greek mathematician Nicomedes.{{cite EB1911|wstitle=Conchoid|volume=6|pages=826–827}}

Description

For every line through {{mvar|O}} that intersects the given curve at {{mvar|A}} the two points on the line which are {{mvar|d}} from {{mvar|A}} are on the conchoid. The conchoid is, therefore, the cissoid of the given curve and a circle of radius {{mvar|d}} and center {{mvar|O}}. They are called conchoids because the shape of their outer branches resembles conch shells.

The simplest expression uses polar coordinates with {{mvar|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 {{math|1=x = a}}, then the line's polar form is {{math|1=r = a sec θ}} 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

References

{{Reflist}}

  • {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=[https://archive.org/details/catalogofspecial00lawr/page/36 36, 49–51, 113, 137] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/36 }}

External links

{{cc}}

  • [https://www.geogebra.org/m/u27bvtre conchoid with conic sections] - interactive illustration
  • {{MathWorld |id=ConchoidofNicomedes |title=Conchoid of Nicomedes}}
  • [https://mathcurve.com/courbes2d.gb/conchoiddenicomede/conchoiddenicomede.shtml conchoid] at mathcurves.com

Category:Plane curves

Category:Greek mathematics

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