Conchoid of de Sluze

The Conchoid of de Sluze for several values of a

The conchoid(s) of de Sluze is a family of plane curves studied in 1662 by René François Walter, baron de Sluze.^[1]^[2]

The curves are defined by the polar equation

r=\sec \theta +a\cos \theta \,

.

(x-1)(x^{2}+y^{2})=ax^{2}\,

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 $a\geq -1$ ,

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

while for $a<-1$ , the area is

\left(1-{\frac a2}\right){\sqrt {-(a+1)}}-a\left(2+{\frac a2}\right)\arcsin {\frac 1{{\sqrt {-a}}}}.

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

\left(2+{\frac a2}\right)a\arccos {\frac 1{{\sqrt {-a}}}}+\left(1-{\frac a2}\right){\sqrt {-(a+1)}}.

Four of the family have names of their own:

a=0, line (asymptote to the rest of the family)

References