NEWTON TRIDENT

 Curve studied by Newton in 1701. Other name: Cartesian parabola. Isaac Newton (1642-1727): English physicist, mathematician, and astronomer.

 Cartesian equation:  where P is a polynomial of degree 3 with valuation 0 Rational cubic with a double point (at infinity in the direction of Oy), the parabola  and the hyperbola  are asymptotes. Reduced Cartesian equation:  (case k = 1 opposite).

 The Newton tridents are defined by the above Cartesian equation; they can be seen as the medians along Oy of the parabolas  and the hyperbolas . The trident as the median of a parabola and a hyperbola. They also appear as contour lines of the cubic surface .

They can also be obtained by hyperbolism from the cubical parabolas.

 Their images by the homographic transformation  are the curves  which have their double point at O. For example, the trident  is transformed into a Cartesian folium. In the opposite figure, we used  instead of  for the sake of readability. The Cartesian folium is a perspective of the Newton trident.