Folium of Descartes
Classic Cartesian curvesThe looped cubic x³ + y³ = 3axy that sparked the Descartes–Fermat duel over tangent lines.
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Equation
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What is the Folium of Descartes?
Descartes proposed the curve x³ + y³ = 3axy in 1638 and challenged Fermat to find its tangent lines — expecting the problem to embarrass him. Fermat’s method of adequality handled it easily, an early landmark on the road to derivatives. The name folium is Latin for “leaf”: the curve loops through the origin in the first quadrant and runs off along two arms.
The folium is the standard first example of implicit differentiation with a genuine payoff: the loop closes at a node where the curve crosses itself with two distinct tangents, and both arms approach the same asymptote x + y + a = 0. Its symmetry axis is the line y = x, where the loop peaks at (3a/2, 3a/2).
Key properties
- Node at the origin with tangents along both axes.
- Asymptote: x + y + a = 0 for both arms.
- The loop’s area is 3a²/2 — and equals the area between the arms and the asymptote.
- Symmetric about y = x; the loop’s vertex is (3a/2, 3a/2).
- Rational parametrization x = 3at/(1+t³), y = 3at²/(1+t³) covers the whole curve.
Where you'll see it
Implicit differentiation lessons, the history of tangent methods before calculus, and algebraic curve galleries.