Conchoid of de Sluze

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The Conchoid of de Sluze for several values of a

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


In cartesian coordinates, the curves satisfy the implicit equation

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 ,

while for , the area is

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

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


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