Folium of Descartes
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 .
The 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 Descartes was unable to do. Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.
Since the equation is degree 3 in both x and y, and does not factor, it is difficult to solve for one of the variables.
However, the equation in polar coordinates is:
which can be plotted easily. By using this formula, the area of the interior of the loop is found to be .
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