Folium of Descartes

A plane algebraic curve of order three which is given in Cartesian coordinates by the equation $ x ^ {3} + y ^ {3} - 3axy = 0 $; the parametric equations are

$$ x = \frac{3at}{1 + t ^ {3} } ,\ y = \frac{3a t ^ {2} }{1 + t ^ {3} } , $$

where $ t $ is the tangent of the angle between the radius vector of the curve and the $ x $- axis. The folium of Descartes is symmetric about the axis $y=x$ (see Fig.). The tangent lines are parallel to the coordinate axes at the points with coordinates $ ( a 2 ^ {1/3} , a 4 ^ {1/3} ) $ and $ ( a 4 ^ {1/3} , a 2 ^ {1/3} ) $. The coordinate origin is a nodal point with the coordinate axes as tangent lines. The asymptote is given by $ y= - x- a $. The surface area enclosed between the curve and the asymptote is $ S = 3a ^ {2} /2 $. The surface area of the loop is $ S = 3a ^ {2} /2 $.

This curve is named after R. Descartes who was the first to study it in 1638.

References

[1]	A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[2]	A.S. Smogorzhevskii, E.S. Stolova, "Handbook of the theory of planar curves of the third order" , Moscow (1961) (In Russian)
[a1]	J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
[a2]	K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)