CUBICAL HYPERBOLA

 red: elliptic cubic with an oval  green: acnodal cubic  blue: elliptic cubic with a branch  yellow: crunodal cubic  magenta: cuspidal cubic

 Curve studied by Newton in 1701. Other names: ambigene hyperbola (name given by Newton), half-trident.

 Cartesian equation:  where P is a polynomial of degree less than or equal to 3 with P(0) = 0. Cubic.

The homographic transformation:  reduces this cubic to the right divergent parabola.
Like the divergent parabolas (as well as the Chasles cubics), the cubical hyperbolas represent the perspective views of all cubics.

 When P is of degree 3, the cubical hyperbola is rational iff P has a multiple root. If, additionally, the leading coefficient is negative, then it can be constructed as a Rosillo curve.  Remarkable cases:  the cissoid of Diocles () and the right strophoid (). case where the leading curve is positive and a root has multiplicity 3: Remarkable cubical hyperbolas in the case where P is of degree 3 and has simple roots:  the Lamé cubic (, equivalent up to bidilatation to the curve opposite), and the Humbert cubic (). (cf the witch of Agnesi)

When P is of degree 2, we get the Hügelschäffer eggs.
When P is of degree 1, we get the witch of Agnesi () or the yellow curve above.

 Case where P is of degree 0

© Robert FERRÉOL  2017