In Geometry, the Folium of Descartes is an algebraic curve defined by the equation
It forms a loop in the first quadrant with a double point
at the origin and asymptote
It is symmetrical about
Then name comes from the Latin word folium which means "leaf".
The curve was featured, along with a portrait of Descartes, on an Albanian stamp in 1966.
The curve was first proposed by Descartes
in 1638. Its claim to fame lies in an incident in the development of calculus
. Descartes challenged 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 the 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
Since the equation is degree 3 in both x and y, and does not factor, it is difficult to find solve for one of the variables. However, the equation in polar coordinates
which can be plotted easily. Another technique is to write y = px and solve for x and y in terms of p. This yields the parametric equations
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
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.