The folium of Descartes

Q: What is the folium of Descartes?

The cubic curve x³ + y³ = 3axy, with a loop in the first quadrant and the asymptote x + y + a = 0.

Q: Why is it called a folium?

Folium is Latin for leaf, after the shape of its loop.

The folium of Descartes is a cubic curve with a leaf-shaped loop in the first quadrant and two arms that run off toward a slanted asymptote. René Descartes proposed it in 1638 to challenge Pierre de Fermat’s method of finding tangents.

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x³ + y³ = 6xy (a = 2) with its asymptote x + y = −2.

Equations to type

Implicit (a = 2)

x^3 + y^3 = 6x y

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Parametric

(6t/(1 + t^3), 6t^2/(1 + t^3))

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At a glance

Equation: x³ + y³ = 3axy
Asymptote: x + y + a = 0
Area of the loop: 3a²/2
Symmetry: About the line y = x

A rational parametrisation

Substituting y = tx turns the cubic into x³(1 + t³) = 3atx², so x = 3at/(1 + t³) and y = 3at²/(1 + t³). The loop is traced for t ≥ 0, and the two arms for t < 0, with t = −1 sending the point off to infinity along the asymptote.

The loop and the node

The curve crosses itself at the origin, a node where two branches meet at right angles along the axes. The loop encloses area 3a²/2, which, remarkably, equals the area between the arms and the asymptote. The tip of the leaf is at (3a/2, 3a/2) on the line y = x.

Tangents by implicit differentiation

Descartes’ challenge was to find tangents to this curve, which cannot be written as y = f(x). Differentiating both sides of x³ + y³ = 3axy with respect to x gives 3x² + 3y²y′ = 3a(y + xy′), so

y′ = (ay − x²) / (y² − ax).

At the tip (3a/2, 3a/2) the slope is −1, perpendicular to the line of symmetry, as it must be. Fermat’s method handled the curve without difficulty, which settled the argument in his favour.

Descartes versus Fermat

In 1637 Descartes published his geometry, with its own way of finding tangents, and Fermat had a different method circulating in letters. Descartes, sceptical of Fermat’s approach, sent the folium as a test case. Both methods are ancestors of the derivative; Fermat’s, which looks at what happens as a small increment shrinks to nothing, is the closer of the two.