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Concept# Conchoid of de Sluze

Summary

In algebraic geometry, the conchoids of de Sluze are a family of plane curves studied in 1662 by Walloon mathematician René François Walter, baron de Sluze.
The curves are defined by the polar equation
:r=\sec\theta+a\cos\theta ,.
In cartesian coordinates, the curves satisfy the implicit equation
:(x-1)(x^2+y^2)=ax^2 ,
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,
:|a|(1+a/4)\pi ,
while for a < −1, the area is
:\left(1-\frac a2\right)\sqrt{-(a+1)}-a\left(2+\frac a2\right)\arcsin\frac1{\sqrt{-a}}.
If a < −1, the curve will have a loop. The area of the loop is
:\left(2+\frac a2\right)a\arccos\frac1{\sq

Official source

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