Conchoid of de Sluze

The Conchoid of de Sluze for several values of Template:Mvar

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.

The curves are defined by the polar equation

$r=\sec \theta +a\cos \theta .$

In cartesian coordinates, the curves satisfy the implicit equation

$(x-1)(x^2+y^2)=a{x}^2$

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,

$|a|(1+a/4)\pi$

while for Template:Math, the area is

$(1-\frac{a}{2})\sqrt{-(a+1)}-a(2+\frac{a}{2})\arcsin \frac{1}{\sqrt{-a}}.$

If Template:Math, the curve will have a loop. The area of the loop is

$(2+\frac{a}{2})a\arccos \frac{1}{\sqrt{-a}}+(1-\frac{a}{2})\sqrt{-(a+1)}.$

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