Folium of Descartes

Summary: The cubic curve $y^3+z^3-cyz=0$ admits the rational parametrization $z=\frac{cx}{1+x^3}$, $y=\frac{c{x}^2}{1+x^3}$, derived by Euler in §52 as an application of the $y=x^m{z}^n$ substitution.

Sources: chapter3

Last updated: 2026-04-23

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Setup

Euler writes the curve as

$y^3+z^3-cyz=0$

and identifies it with his general §52 template $a{y}^{\alpha }+b{z}^{\beta }+c{y}^{\gamma }{z}^{\delta }=0$ by setting

$a=-1,b=-1,\alpha =3,\beta =3,\gamma =1,\delta =1$

(source: chapter3, §52 example).

Parametrization by Method I

Method I of §52 chooses $n=\beta /\alpha =1$, so $y=x^{m}z$. Taking $m=1$:

$y=xz.$

Substituting into $y^3+z^3-cyz=0$:

$x^3{z}^3+z^3-cx{z}^2=0⟺z^2(z(x^3+1)-cx)=0,$

giving $z=\frac{cx}{1+x^3}$ and hence

$\boxed{z=\frac{cx}{1+x^3},y=\frac{c{x}^2}{1+x^3}.}$

Both are rational functions of $x$ (source: chapter3, §52 example).

The other two methods

Euler also records the two alternative choices of $n$ from §52:

• Method II ($n=(\beta -\delta )/\gamma =2$) gives $z=\frac{1}{x}(cx-1{)}^{1/3}$ and $y=\frac{1}{x}(cx-1{)}^{2/3}$.
• Method III ($n=\delta /(\alpha -\gamma )=1/2$, with $m=1$ in the formula as shown) gives $z=(cx-x^3{)}^{2/3}$ and $y=x(cx-x^3{)}^{1/3}$.

Only Method I produces a rational parametrization; the others are “radical but single-valued in $x$” forms (source: chapter3, §52 example).

Geometric interpretation

The folium of Descartes has a node at the origin. The substitution $y=xz$ is geometrically projection from the node: $x=y/z$ is the slope of the line through the origin, and each non-origin point of the curve lies on a unique such line. This is the same projection-from-a-special-point trick that produces the rational parametrization of the circle from the point $(-a,0)$, but applied to a singular point rather than an ordinary one.

In modern language, the folium is a rational (genus-zero) cubic, and §52 Method I is explicitly its rational parametrization.

Related pages

• homogeneous-substitution
• substitution
• chapter-3-on-the-transformation-of-functions-by-substitution
• rational-parametrization-of-the-circle