Folium of Descartes

In geometry, the folium of Descartes is an algebraic curve defined by the equation 
The name comes from the Latin word "folium" which means "leaf".
History.
The curve was first proposed and studied by René Descartes in 1638. Its claim to fame lies in an incident in the development of calculus. Descartes challenged Pierre de Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do. Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.
Graphing the curve.
The folium of Descartes can be expressed in polar coordinates as 
which is plotted on the left. This is equivalent to
formula_3
Another technique is to write formula_4 and solve for formula_5 and formula_6 in terms of formula_7. This yields the rational parametric equations:
formula_8.
We can see that the parameter is related to the position on the curve as follows:
Another way of plotting the function can be derived from symmetry over formula_18. The symmetry can be seen directly from its equation (x and y can be interchanged). By applying rotation of 45° CW for example, one can plot the function symmetric over rotated x axis.
This operation is equivalent to a substitution:
and yields
Plotting in the Cartesian system of formula_21 gives the folium rotated by 45° and therefore symmetric by formula_22-axis.
Properties.
It forms a loop in the first quadrant with a double point at the origin and asymptote
It is symmetrical about the line formula_18. As such, the two intersect at the origin and at the point formula_25.
Implicit differentiation gives the formula for the slope of the tangent line to this curve to beformula_26Using either one of the polar representations above, the area of the interior of the loop is found to be formula_27. Moreover, the area between the "wings" of the curve and its slanted asymptote is also formula_28.
Relationship to the trisectrix of Maclaurin.
The folium of Descartes is related to the trisectrix of Maclaurin by affine transformation. To see this, start with the equation 
and change variables to find the equation in a coordinate system rotated 45 degrees. This amounts to setting 
In the formula_31 plane the equation is 
If we stretch the curve in the formula_33 direction by a factor of formula_34 this becomes 
which is the equation of the trisectrix of Maclaurin.