Conchoid (mathematics)
Template:Legend Template:Legend-line Each pair of coloured curves is length Template:Mvar from the intersection with the line that a ray through Template:Mvar makes. Template:Legend-line Template:Legend-line Template:Legend-line
In geometry, a conchoid is a curve derived from a fixed point Template:Mvar, another curve, and a length Template:Mvar. It was invented by the ancient Greek mathematician Nicomedes.<ref>Template:Cite EB1911</ref>
DescriptionEdit
For every line through Template:Mvar that intersects the given curve at Template:Mvar the two points on the line which are Template:Mvar from Template:Mvar are on the conchoid. The conchoid is, therefore, the cissoid of the given curve and a circle of radius Template:Mvar and center Template:Mvar. They are called conchoids because the shape of their outer branches resembles conch shells.
The simplest expression uses polar coordinates with Template:Mvar 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, then the conchoid is the conchoid of Nicomedes.
For instance, if the curve is the line Template:Math, then the line's polar form is Template:Math and therefore the conchoid can be expressed 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 alsoEdit
ReferencesEdit
External linksEdit
- conchoid with conic sections - interactive illustration
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:ConchoidofNicomedes%7CConchoidofNicomedes.html}} |title = Conchoid of Nicomedes |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}
- conchoid at mathcurves.com