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.[1][2]

The Conchoid of de Sluze for several values of a

The curves are defined by the polar equation

${\displaystyle r=\sec \theta +a\cos \theta \,}$.

In cartesian coordinates, the curves satisfy the implicit equation

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

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 ${\displaystyle a\geq -1}$,

${\displaystyle |a|(1+a/4)\pi \,}$

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

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

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

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

Four of the family have names of their own:

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

a=−1, cissoid of Diocles

a=−2, right strophoid

a=−4, trisectrix of Maclaurin

References

1. Smith, David Eugene (1958), History of Mathematics, Volume 2, Courier Dover Publications, p. 327, ISBN 9780486204307.
2. "Conchoid of de Sluze by J. Dziok et al.on Computers and Mathematics with Applications 61 (2011) 2605–2613" (PDF).