next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
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 (). 

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.


next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL 2017