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.

In polar coördinates

<math>r=\sec\theta+a\cos\theta<math>

and in implicit Cartesian coördinates

<math>(x-1)(x^2+y^2)=ax^2<math>

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

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

<math>|a|(1+a/4)\pi<math> for a≥−1

<math>\left(1-\frac a2\right)\sqrt{-(a+1)}-a\left(2+\frac a2\right)\arcsin\frac1{\sqrt{-a}}<math> for a<−1

The area of the loop is

<math>\left(2+\frac a2\right)a\arccos\frac1{\sqrt{-a}}

+ \left(1-\frac a2\right)\sqrt{-(a+1)}<math> for a<−1

Four of the family have names of their own:

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

External links

• Mathworld
• 2D Curves
• Famous Curves (includes a scaling factor)