# .

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

The curves are defined by the polar equation

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

In cartesian coordinates, the curves satisfy the implicit equation

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

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 $$a \ge -1,$$

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

while for a < -1, the area is

$$\left(1-\frac a2\right)\sqrt{-(a+1)}-a\left(2+\frac a2\right)\arcsin\frac1{\sqrt{-a}}.$$

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

$$\left(2+\frac a2\right)a\arccos\frac1{\sqrt{-a}} + \left(1-\frac a2\right)\sqrt{-(a+1)}.$$

Four of the family have names of their own:

a=0, line (asymptote to the rest of the family)
a=−1, cissoid of Diocles
a=−2, right strophoid
a=−4, trisectrix of Maclaurin

References

Smith, David Eugene (1958), History of Mathematics, Volume 2, Courier Dover Publications, p. 327, ISBN 9780486204307.