Conchoid of de Sluze

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The Conchoid of de Sluze for varying values of a
The Conchoid of de Sluze for varying 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.

In polar coordinates

r = secθ + acosθ

and in implicit Cartesian coordinates

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

The area of the loop is

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

Four of the family have names of their own:

[edit] See also

[edit] External links

In other languages