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CARTESIAN FOLIUM

Curve studied by Descartes and Roberval in 1638 then by Huygens in 1672.
From the Latin folium "leaf".
René Descartes (1596-1650): French philosopher, mathematician and physicist. Other name, given by Roberval: jasmine flower. |

Cartesian equation: .
Polar equation: . Cartesian parametrization: . Rational cubic with a double point. The area of the loop is equal to that of the domain located between the curve and its asymptote (of equation |

The Cartesian folium is, in general, not defined by a geometrical property, but by its Cartesian equation, given above.

The Cartesian equation in a frame turned by p/4 with respect to the previous one is:
where b = ,
equation to relate to that of the Maclaurin trisectrix: .
Therefore, the Cartesian folium is none other than the image of this trisectrix by a dilatation of the axis *Ox* by a ratio .

The Cartesian folium is therefore also a cissoid of an ellipse and a line (images by the above transformation of the circle and the line associated to the Maclaurin trisectrix), so a cissoid of Zahradnik.

See also trident of Newton.

The cubic surface z = x^{3}
+ y^{3} -3xy
cut by the plane z = 0.... |

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© Robert FERRÉOL, Jacques MANDONNET 2017