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Folium of Descartes — Definition, Formula & Examples

The Folium of Descartes is a loop-shaped algebraic curve defined by the equation x3+y3=3axyx^3 + y^3 = 3axy, where aa is a positive constant that controls the size of the loop. It was first discussed by René Descartes in 1638 and is frequently used in calculus courses to practice implicit differentiation and parametric equations.

The Folium of Descartes is the plane algebraic curve given implicitly by x3+y3−3axy=0x^3 + y^3 - 3axy = 0 for a parameter a>0a > 0. It admits the rational parametrization x=3at1+t3x = \frac{3at}{1+t^3}, y=3at21+t3y = \frac{3at^2}{1+t^3} for t≠−1t \neq -1, passes through the origin (where it has a node), and possesses the asymptote x+y+a=0x + y + a = 0.

Key Formula

x3+y3=3axyx^3 + y^3 = 3axy
Where:
  • xx = Horizontal coordinate
  • yy = Vertical coordinate
  • aa = Positive constant controlling the size of the loop

How It Works

The curve forms a single loop in the first quadrant, crosses itself at the origin, and extends into the third quadrant along two branches that approach the slant asymptote x+y=−ax + y = -a. To find dydx\frac{dy}{dx}, you apply implicit differentiation to x3+y3=3axyx^3 + y^3 = 3axy. The parametric form with parameter t=y/xt = y/x is especially useful for computing arc length or enclosed area, since it converts the implicit equation into rational expressions in tt. The loop encloses a finite area equal to 3a22\frac{3a^2}{2}.

Worked Example

Problem: Find dy/dx for the Folium of Descartes with a = 1, given by x³ + y³ = 3xy. Evaluate the slope at the point (3/2, 3/2).
Differentiate implicitly: Differentiate both sides with respect to x.
3x2+3y2dydx=3y+3xdydx3x^2 + 3y^2 \frac{dy}{dx} = 3y + 3x\frac{dy}{dx}
Solve for dy/dx: Collect terms with dy/dx on one side and factor.
dydx=y−x2y2−x\frac{dy}{dx} = \frac{y - x^2}{y^2 - x}
Substitute the point: Plug in x = 3/2 and y = 3/2.
dydx=32−9494−32=−3434=−1\frac{dy}{dx} = \frac{\frac{3}{2} - \frac{9}{4}}{\frac{9}{4} - \frac{3}{2}} = \frac{-\frac{3}{4}}{\frac{3}{4}} = -1
Answer: The slope at (3/2, 3/2) is −1-1, confirming the curve is symmetric about the line y = x (the tangent is perpendicular to this line at the tip of the loop).

Why It Matters

The Folium of Descartes is a standard exercise in multivariable and single-variable calculus for practicing implicit differentiation, parametric integration, and finding tangent lines at singular points. It also appears in algebraic geometry as a simple example of a rational nodal curve.

Common Mistakes

Mistake: Forgetting the node at the origin when computing tangent lines.
Correction: At the origin, the curve crosses itself, so there are two tangent lines (y = 0 and x = 0), not one. Standard implicit differentiation gives 0/0 there; you need to factor or use the parametric form instead.