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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), halftrident. 
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.


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© Robert FERRÉOL 2017