The conchoid of de Sluze is the cubic curve first constructed by René de Sluze in 1662. It is given by the implicit equation
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or the polar equation
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This can be written in parametric form as
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The conchoid of de Sluze has a singular point at the origin which is a crunode for 
, a cusp for 
, and an acnode for 
.
It has curvature and tangential angle
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![(2a(4+a-3sec^2t))/([a(4+a)-2asec^2t+sec^4t]^(3/2))](http://mathworld.wolfram.com/images/equations/ConchoidofdeSluze/Inline12.gif) |
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![2t-tan^(-1)[(2sin(2t))/(2+(a+2)cos(2t))].](http://mathworld.wolfram.com/images/equations/ConchoidofdeSluze/Inline15.gif) |
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The curve has a loop if 
, in which case the loop is swept out by 
. The area of the loop is
![A_(loop)=1/2[(2-a)sqrt(-a-1)+a(4+a)sec^(-1)(sqrt(-a))].](http://mathworld.wolfram.com/images/equations/ConchoidofdeSluze/NumberedEquation3.gif) |
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SEE ALSO:Conchoid, Conchoid of Nicomedes
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.
