Conchoid of de Sluze
In
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 a ≥ −1,
while for a < −1, the area is
If a < −1, 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