Conchoid of de Sluze

The conchoid(s) of de Sluze is a family of plane curves studied in 1662 by René François Walter, baron de Sluze.

The curves are defined by the polar equation

In cartesian coordinates, the curves satisfy the implicit equation

except that for a=0 the implicit form has an acnode (0,0) not present in polar form.

They are rational, circular, cubic plane curves.

These expressions have an asymptote x=1 (for a≠0). The point most distant from the asymptote is (1+a,0). (0,0) is a crunode for a<−1.

The area between the curve and the asymptote is, for ${\displaystyle a\geq -1}$,

while for ${\displaystyle a<-1}$, the area is

If ${\displaystyle a<-1}$, the curve will have a loop. The area of the loop is

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