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DUPLICATRIX CUBIC

Curve studied by Ullhorn in 1809, and G. de Longchamps
in 1890.
Other name: toxoid. |

View with the asymptotic cubic (in green). |
Polar equation:
.
Cartesian equation: , i.e. . Cartesian parametrization: . Rational cubic (even polynomial) with an isolated point (0,0), that is not obtained with the polar equation. |

Given two perpendicular lines *D*_{1}
(here *Ox*) and *D*_{2} (here *x*
= *a*) and a point *O* on *D*_{1},
the duplicatrix cubic is the locus of the point
*M* on a variable
line *D* passing through *O* such that the projection on *D*
of the projection of *M* on *D*_{1}
is a point on *D*_{2}.

This curve is a special case of divergent parabola and of Clairaut's curve.

It is also the inverse of the simple folium with respect to its "pointed" summit,

As one can tell by its name, it is a duplicatrix:
indeed, when
, .

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© Robert FERRÉOL 2017