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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), 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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© Robert FERRÉOL  2017