Lemniscate of Bernoulli
Polar curvesThe figure-eight curve r² = a²·cos 2θ — the set of points whose distances to two foci multiply to a constant.
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Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Lemniscate of Bernoulli?
The lemniscate of Bernoulli is the locus of points whose distances to two foci have a constant product — the multiplicative cousin of the ellipse, which fixes the sum. Jacob Bernoulli described it in 1694 and named it after the Latin lemniscus, a hanging ribbon. Its polar form r² = a²·cos 2θ shows why the curve only exists where cos 2θ ≥ 0: two symmetric lobes meeting at the origin.
The curve became far more than a pretty figure eight: attempts to compute its arc length led Fagnano and Euler to the lemniscatic integrals, a direct ancestor of the theory of elliptic functions.
Key properties
- Total enclosed area: a² (both lobes together).
- Foci sit at (±a/√2, 0); the distance product equals a²/2.
- The origin is a crossing point where the two tangents are y = ±x.
- It is the inverse of the rectangular hyperbola with respect to its center.
- Arc length involves the lemniscate constant ϖ ≈ 2.6221, computed via elliptic integrals.
Where you'll see it
The infinity symbol ∞, analog signal constellations, and the historical road from arc length to elliptic functions.