Conchoid of de Sluze
In algebraic geometry, the conchoids of de Sluze are a family of plane curves studied in 1662 by Walloon mathematician René François Walter, baron de Sluze.<ref>Template:Citation.</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
The curves are defined by the polar equation
- <math>r=\sec\theta+a\cos\theta \,.</math>
In cartesian coordinates, the curves satisfy the implicit equation
- <math>(x-1)(x^2+y^2)=ax^2 \,</math>
except that for Template:Math the implicit form has an acnode Template:Math not present in polar form.
They are rational, circular, cubic plane curves.
These expressions have an asymptote Template:Math (for Template:Math). The point most distant from the asymptote is Template:Math. Template:Math is a crunode for Template:Math.
The area between the curve and the asymptote is, for Template:Math,
- <math>|a|(1+a/4)\pi \,</math>
while for Template:Math, the area is
- <math>\left(1-\frac a2\right)\sqrt{-(a+1)}-a\left(2+\frac a2\right)\arcsin\frac1{\sqrt{-a}}.</math>
If Template:Math, the curve will have a loop. The area of the loop is
- <math>\left(2+\frac a2\right)a\arccos\frac1{\sqrt{-a}} + \left(1-\frac a2\right)\sqrt{-(a+1)}.</math>
Four of the family have names of their own:
- Template:Math, line (asymptote to the rest of the family)
- Template:Math, cissoid of Diocles
- Template:Math, right strophoid
- Template:Math, trisectrix of Maclaurin