Parametric Cartesian equation:
x2y + aby - a2x = 0, ab > 0
Click below to see one of the Associated curves.
|Definitions of the Associated curves||Evolute|
|Involute 1||Involute 2|
|Inverse curve wrt origin||Inverse wrt another circle|
|Pedal curve wrt origin||Pedal wrt another point|
|Negative pedal curve wrt origin||Negative pedal wrt another point|
|Caustic wrt horizontal rays||Caustic curve wrt another point|
The incomparable Sir Isaac Newton gives this following Ennumeration of Geometrical Lines of the Third or Cubick Order; in which you have an admirable account of many Species of Curves which exceed the Conick-Sections, for they go no higher than the Quadratick or Second Order.
[Here the older spelling has been preserved including conick, Quadratick and ennumeration].
Newton shows that the curve f(x, y) = 0, where f(x, y) is a cubic, can be divided into one of four normal forms. The first of these is equations of the form
xy2 + ey = ax3 + bx2 + cx + d.
This is the hardest case in the classification and the serpentine is one of the subcases of this first normal form.
The serpentine had been studied earlier by de L'Hôpital and Huygens in 1692.
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