# Folium of Descartes (Noeud de Ruban)

The folium of Descartes (or noeud de ruban in French) is a single loop with one node (ordinary double point) and two asymptotes at the ends.

It is an algebraic curve defined by the equation x3 + y3 – 3axy = 0.

The curve can be graphed with parametric equations, as shown in this image:

However, the parametric equations are not defined at t = -1. Therefore, you have to create a piecewise function graph that excludes t = 1 (In other words, graph two parts, one either side of t = 1).

## Folium of Descartes Properties

• The curve is symmetric about the line y = x.
• One ordinary double point at the origin.
• Area of loop interior = 3a2/2.
• Area between asymptote and curve “wing” = 3a2/2.
• Horizontal tangent lines at the origin and the point

The curve of Descartes is related to the trisectrix of Maclaurin by affine transformation.

## Finding Tangent Lines

Tangent lines for the curves can be found using implicit differentiation.

Example question: Find the tangent line for the point (2, 4) on Descartes folium.

Solution:

Step 1: Differentiate both sides of the equation with respect to x:

Step 2: Insert the coordinates (2, 4) into the formula and solve:

## History of the Folium of Descartes

The folium of Descartes is named after RenÃ© Descartes (1596 to 1650), who was the first to discuss it. He discovered the folium thanks to a challenge he put out to Fermat — to find a tangent line for an arbitrary point. While Fermat won the challenge, Descartes discovered his folium during the process [1].

The curve provided a role in the early development of calculus and provides the proof for some parts of Fermat’s Last Theorem [2].

Albania issued a postage stamp depicting Descartes and his folium in 1966 [1].

## References

[1] Amoroso, R. FE, FI, FO, FOLIUM: A DISCOURSE ON DESCARTESâ€™ MATHEMATICAL CURIOSITY from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.544.6091&rep=rep1&type=pdf
[2] Pricopie, S. & Udriste, C. Multiplicative group law on the folium of Descartes. Retrieved March 6, 2021 from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.402.4317&rep=rep1&type=pdf