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 x + y = a); common value: 3a2/2.

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. The cubic surface cut by the plane z = 0....