Devil’s curve
The quartic y²(y² − b²) = x²(x² − a²), studied by Cramer in 1750: a figure eight between two arcs, or a circle crossed by two lines.
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Formulas and key values
- Cartesian equation
- Polar equation
- Asymptote
a and b are where the curve meets the axes, at (±a, 0) and (0, ±b); b = 1 in the graph.
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Start the video01 · Definition
What is the Devil’s curve?
Gabriel Cramer studied this quartic in 1750; the name probably comes from its resemblance to the devil-on-two-sticks juggling toy. On the axes it passes through the origin and through (±a, 0) and (0, ±b), and far from the origin it hugs the diagonals y = ±x.
Everything depends on how a compares with b. When a < b, a figure eight crosses itself at the origin between (±a, 0), while two separate arcs pass through (0, ±b) and run out along the diagonals. At a = b the curve falls apart into the circle x² + y² = b² and the two diagonals; when a > b the picture turns through 90°, with the figure eight upright. Dividing out r² gives a one-line polar form that makes each case easy to sketch.
02 · Properties
Key properties
- Crosses the axes at the origin, (±a, 0) and (0, ±b).
- Asymptotes y = ±x.
- a = b: the curve splits into the circle x² + y² = a² and the lines y = ±x.
- Polar form: r² = (b²·sin²θ − a²·cos²θ) / (sin²θ − cos²θ).
- Symmetric about both axes.
03 · Questions
Devil’s curve: common questions
- What is the equation of the Devil’s curve?
- Cartesian equation: y^2(y^2 - b^2) = x^2(x^2 - a^2); Polar equation: r^2 = (b^2 sin^2θ - a^2 cos^2θ)/(sin^2θ - cos^2θ). a and b are where the curve meets the axes, at (±a, 0) and (0, ±b); b = 1 in the graph.
04 · Keep exploring