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

r = secθ + acosθ

and in implicit Cartesian coördinates

(x - 1)(x² + y²) = ax²

(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

| a | (1 + a / 4)π for a≥−1

for a<−1

The area of the loop is

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)