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Conchoid of de Sluze

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ConchoidofdeSluze

The conchoid of de Sluze is the cubic curve first constructed by René de Sluze in 1662. It is given by the implicit equation

(x-1)(x^2+y^2)=ax^2,
(1)

or the polar equation

r=sectheta+acostheta.
(2)

This can be written in parametric form as

x=(sect+acost)cost
(3)
y=(sect+acost)sint.
(4)

The conchoid of de Sluze has a singular point at the origin which is a crunode for a<-1, a cusp for a=-1, and an acnode for a>-1.

It has curvature and tangential angle

kappa(t)=(2a(4+a-3sec^2t))/([a(4+a)-2asec^2t+sec^4t]^(3/2))
(5)
phi(t)=2t-tan^(-1)[(2sin(2t))/(2+(a+2)cos(2t))].
(6)

The curve has a loop if a<-1, in which case the loop is swept out by -sec^(-1)sqrt(-a)<=t<=sec^(-1)sqrt(-a). The area of the loop is

A_(loop)=1/2[(2-a)sqrt(-a-1)+a(4+a)sec^(-1)(sqrt(-a))].
(7)

See also

Conchoid, Conchoid of Nicomedes

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References

MacTutor History of Mathematics Archive. "Conchoid of de Sluze." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Conchoidsl.html.Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, p. 327, 1958.Wassenaar, J. "Conchoid of de Sluze." http://www.2dcurves.com/cubic/cubiccs.html.

Cite this as:

Weisstein, Eric W. "Conchoid of de Sluze." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConchoidofdeSluze.html

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