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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 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.

See also trident of Newton.
 
 
The cubic surface z = x3 + y3 -3xy cut by the plane z = 0....

 
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© Robert FERRÉOL, Jacques MANDONNET 2017