next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

CUBIC

A cubic is an algebraic curve of degree 3.
A cubic always has real points and we will assume that it is not decomposed into a conic and a line.

There exists two large classes of cubics:
    - the elliptic cubics, without singularities (genus 1),
    - the rational cubics, with a singularity (genus 0).

Newton proved that any cubic is projectively equivalent to a divergent parabola, with equation: .

The cubic is then elliptic if and only if  .
If  has three real roots), the curve is composed of an infinite branch and an oval. Example: .

If  has only one real root), the curve is only composed of an infinite branch. Example:  .

In the elliptic case, every value of D gives a projective equivalence class.
In the rational case (), there are three equivalence classes:

 - the crunodal cubics, which have a double point with real tangents, such as  ,or  .

 - the acnodal cubics, which have an isolated double point (imaginary tangents), such as  or .

 - the cuspidal cubics, which have a cuspidal point of the first kind, such as .
 

The divergent parabolas, the cubic hyperbolas, and the cubics of Chasles are families of cubics, and every one of them include these three types of cubics.

See here a website that lists all the cubics linked to the triangle.
 
next curve previous curve 2D curves 3D curves surfaces fractals polyhedra

© Robert FERRÉOL  2017