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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 (D1) and (D2) (here, the lines x = a and y = b), a variable line (D) passing by O meets (D1) at P; the perpendicular at P meets (D2) at Q; the perpendicular at Q meets (D2) 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 D2 passes by O (b = 0), in which case it is symmetrical about (D2). 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