Cubical hyperbola

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

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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)