Definition:Folium of Descartes
Definition
Cartesian Form
The folium of Descartes is the locus of the equation:
- $x^3 + y^3 - 3 a x y = 0$
Parametric Form
The folium of Descartes is the locus of the equation given in parametric form as:
- $\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
Also see
- Results about the folium of Descartes can be found here.
Source of Name
This entry was named for René Descartes.
Historical Note
Marin Mersenne had communicated to René Descartes the method devised by Pierre de Fermat for calculating the tangent to a curve.
René Descartes seems to have thought little of this method, believing that it was not sufficiently general to be useful.
The curve now known as the folium of Descartes was used by him in $1638$ as a challenge to Fermat, believing that he would be unable to use this method on it.
Reportedly he was seriously annoyed at Fermat when the latter solved it without any trouble.
Linguistic Note
The word folium in the term folium of Descartes derives from the Latin for leaf, from the leaf-shaped loop that it encloses.
Sources
- Weisstein, Eric W. "Folium of Descartes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FoliumofDescartes.html