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ANGUINEA
Curve studied by L'Hospital and Huygens in 1692, then
by Newton in 1701.
From the Latin anguis "snake, hydra, dragon" (your choice !), name given by Newton. Other name : serpentine cubic. 

Cartesian equation: .
Cartesian parametrization:.
Polar equation: . 
The anguinea is the hyperbolism of the circle with respect to a point O of this circle and a straight line parallel to the diameter passing through O. Here, the circle is the circle of diameter [OA] with A(a, 0) and the line, y = d. 

Like the witch of Agnesi, it is a projection of the horopter. 

In the figure hereafter, we used instead of for more readibility.
The anguinea is a directrix curve of the Plücker's conoid.
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© Robert FERRÉOL 2016