Conchoid of de Sluze

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.^[1]^[2]

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 $a \neq 0$ ). 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 $a \geq -1$ ,

|a|(1+a/4)\pi \,

while for $a < -1$ , the area is

\left(1-{\frac {a}{2}}\right){\sqrt {-(a+1)}}-a\left(2+{\frac {a}{2}}\right)\arcsin {\frac {1}{\sqrt {-a}}}.

If $a < -1$ , the curve will have a loop. The area of the loop is

\left(2+{\frac {a}{2}}\right)a\arccos {\frac {1}{\sqrt {-a}}}+\left(1-{\frac {a}{2}}\right){\sqrt {-(a+1)}}.

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

References

^ Smith, David Eugene (1958), History of Mathematics, Volume 2, Courier Dover Publications, p. 327, ISBN 9780486204307.
^ "Conchoid of de Sluze by J. Dziok et al.on Computers and Mathematics with Applications 61 (2011) 2605–2613" (PDF).

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

[2] "Conchoid of de Sluze by J. Dziok et al.on Computers and Mathematics with Applications 61 (2011) 2605–2613" (PDF).

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