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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