View source for 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.'''<br/>
{{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|thumb|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 {{mvar|O}}, another curve, and a length {{mvar|d}}. It was invented by the ancient Greek mathematician [[Nicomedes (mathematician)|Nicomedes]].<ref>{{cite EB1911|wstitle=Conchoid|volume=6|pages=826–827}}</ref>

==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|conch shells]].

The simplest expression uses polar coordinates with {{mvar|O}} at the origin. If 
:<math>r=\alpha(\theta)</math>
expresses the given curve, then 
:<math>r=\alpha(\theta)\pm d </math>
expresses the conchoid. 

If the curve is a [[Line (mathematics)|line]], then the conchoid is the ''conchoid of [[Nicomedes (mathematician)|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 [[Parametric_equation|parametrically]] as 
:<math>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.

==See also==
*[[Cissoid]]
*[[Strophoid]]

==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]]
{{geometry-stub}}