Folium of Descartes

In geometry, the folium of Descartes (from Latin folium 'leaf'; named for René Descartes) is an algebraic curve defined by the implicit equation

The curve was first proposed and studied by René Descartes in 1638.

[1] 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.

[2] Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.

[3] The folium of Descartes can be expressed in polar coordinates as

which is plotted on the left.

Another technique is to write

This yields the rational parametric equations:[5] 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

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:

Plotting in the Cartesian system of

gives the folium rotated by 45° and therefore symmetric by

It forms a loop in the first quadrant with a double point at the origin and asymptote

It is symmetrical about the line

As such, the two intersect at the origin and at the point

Implicit differentiation gives the formula for the slope of the tangent line to this curve to be[3]

{\displaystyle {\frac {dy}{dx}}={\frac {ay-x^{2}}{y^{2}-ax}}.}

Using either one of the polar representations above, the area of the interior of the loop is found to be

Moreover, the area between the "wings" of the curve and its slanted asymptote is also

[1] 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

plane the equation is

If we stretch the curve in the

direction by a factor of

which is the equation of the trisectrix of Maclaurin.