Conchoid of de Sluze



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

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 $a &ne; 0$). The point most distant from the asymptote is $(1 + a, 0)$. $(0,0)$ is a crunode for $a < &minus;1$.

The area between the curve and the asymptote is, for $a &ge; &minus;1$,
 * $$|a|(1+a/4)\pi \,$$

while for $a < &minus;1$, the area is
 * $$\left(1-\frac a2\right)\sqrt{-(a+1)}-a\left(2+\frac a2\right)\arcsin\frac1{\sqrt{-a}}.$$

If $a < &minus;1$, 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)
 * $a = &minus;1$, cissoid of Diocles
 * $a = &minus;2$, right strophoid
 * $a = &minus;4$, trisectrix of Maclaurin