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PARABOLIC FOLIUM

Curve studied by Gérard de Longchamps in 1890. |

Cartesian equation: .
Polar equation: . Rational cubic with a double point. |

Given a point *O* and two perpendicular lines (*D*_{1}) and (*D*_{2}) (here, the lines *x* = *a* and *y* = *b*), a variable line (*D*) passing by *O* meets
(*D*_{1}) at *P*; the perpendicular at *P* meets (*D*_{2}) at *Q*; the perpendicular at *Q* meets (*D*_{2}) at *R*; the perpendicular at *R* meets (*D*) at *M*: the *parabolic folium* (which is not a folium) is the locus of *M*.

The parabolic folium is said right when D_{2} passes by O (b = 0), in which case it is symmetrical about (D_{2}). Therefore, it is a special case of divergent parabola and of tear drop.
Its equation, in the form , shows that it is the cissoid of a line and a semicubical parabola. |

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© Robert FERRÉOL, Jacques MANDONNET 2017