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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:.
Rational cubic with isolated point (at infinity in the direction of Oy).

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.
To bring back the isolated point at infinty to a finite distance, we can use the homographic transformation  :  that transforms the anguinea  into the mixed cubic having its isolated point in O.

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