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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.

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© Robert FERRÉOL 2017