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.

 

Cartesian equation:  where P is a polynomial of degree 3. Cubic.

The divergent parabolas are the curves defined by the above Cartesian equation.

Any cubic is projectively equivalent to (i.e. is the image by a real homographic transformation of) a divergent parabola (Newton theorem). Since all homographies are composed of a direct isometry and a homology (or perspective), this means that any cubic can be seen under a certain perspective as a right divergent parabola.

Therefore, the cubic is rational if and only if the discriminant D of the polynomial P above is equal to 0, and, in this case, there are three projective equivalence classes, composed of the crunodal, acnodal and cuspidal cubics.
In the non-rational case (cubic of genus 1, called "elliptic cubic"), every value of D gives a projective equivalence class.
 
 

Arrangement of the zeros of P	Type of the cubic
Three distinct real zeros; reduced equation:	Elliptic cubic with an oval
A real zero and two conjugate complex zeros; reduced equation:	Elliptic cubic with a branch
Two real zeros, one of which has multiplicity 2, and P(x) ³ 0 in a neighbourhood of the double root; Reduced Cartesian equation: . Cartesian parametrization: . Therefore, the cubic is polynomial. This case includes the Tschirnhausen cubic: (b = 3 a), the right parabolic folium (b = a) and a case of Lissajous curve (a = 3 b).	Crunodal cubic (with double point)
Two real zeros, one of which has multiplicity 2, and P(x) £ 0 in the neighbourhood of the double root; Reduced equation . Cartesian parametrization: . Polynomial cubic. This case includes the duplicatrix cubic (b = a).	Acnodal cubic (with an isolated point)
A triple real zero: semicubical parabola.	Cuspidal cubic (with a cuspidal point)

In the last three cases, the divergent parabola can be obtained by antihyperbolism of the parabola (hence the name divergent parabola).

See also the Chasles cubics, the cubical hyperbolas (other families of curves representing all cubics), the pursuit curves, and the swimming dog curve.
 
 

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© Robert FERRÉOL 2017