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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 2017