Cartesian equation: xy = cx
+dx
+ex+f
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 curve occurs in Newton's study of cubics. It is contained in his classification of cubic curves which appears in Curves by Sir Isaac Newton in Lexicon Technicum by John Harris published in London in 1710.
Newton was the first to undertake such a systematic study of cubic equations and he classified them into 72 different cases. In fact he missed six cases in his classification. The trident is the sixty sixth species in his classification and Newton gives the graph essentially looking identical to the graph givenabove.
Newton states some properties of his trident. For example he states that the curve has four infinite legs and that the y-axis is an asymptote to two tending towards contrary parts . He also comments And this figure is that parabola by which D. Cartes (Descartes) constructed equations of six dimensions .
Newton's classification of cubics was criticised by Euler because it lacked general general principle. Plücker later gave a more detailed classification with 219 types.
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