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DIVERGENT PARABOLA

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, Jacques MANDONNET 2017