What Is a Lemniscate?
A lemniscate is a graceful figure‑8 or infinity‑symbol curve that arises from a beautifully simple rule: it is the set of all points P such that the product of P's distances from two fixed foci equals a constant. Think of it as the "multiplicative" cousin of the ellipse — where an ellipse keeps the sum of distances constant, the lemniscate keeps the product constant.
A Name With History
Swiss mathematician Jakob Bernoulli first described this curve in 1694 in a paper published in Acta Eruditorum, the leading scientific journal of the day. While investigating the shape taken by an elastic band under certain loads, he encountered the curve and named it the lemniscate.
The word comes from the Latin lemniscus — a ribbon or pendant — which itself derives from the Greek lemniskos, the woolen ribbons tied to victor's garlands in ancient athletic competitions. Bernoulli chose the name because the curve's two interlocking lobes suggested a ribbon tied in a figure‑8 bow.
Did you know? The ∞ symbol was introduced by John Wallis in 1655 — nearly 40 years before Bernoulli described the lemniscate — so the infinity symbol was not named after the curve. However the two are so visually similar that they are almost always associated with each other.
The curve later attracted the attention of Leonhard Euler (1750s), who discovered that computing its arc length leads to an integral that cannot be expressed using ordinary functions. This launched the study of elliptic integrals and eventually elliptic functions, a major branch of 19th–century mathematics developed by Gauss, Abel, and Jacobi. The lemniscate constant ω ≈ 2.6221 plays the same role for the lemniscate that π plays for the circle.
The Polar Equation
The lemniscate is most elegantly written in polar coordinates as:
r² = a² · cos(2θ)
where r is the distance from the origin, θ is the angle, and a controls the size (the half‑width of each lobe equals a). Note that this equation only has solutions when cos(2θ) ≥ 0 — the curve simply does not exist for angles outside those two arcs. That is what creates the two separate lobes.
The equivalent Cartesian equation is:
(x² + y²)² = a² · (x² − y²)
From Polar to Parametric
To draw the curve smoothly with code, we convert to parametric equations — functions x(t), y(t) that trace the entire figure‑8 in one smooth pass for t ∈ [0, 2π):
x(t) = a · cos(t) / (1 + sin²(t))
y(t) = a · sin(t) · cos(t) / (1 + sin²(t))
where a is the Scale parameter.
Verification — these equations satisfy (x² + y²)² = a²(x² − y²):
Let d = 1 + sin²(t). Then x = a cos(t)/d and y = a sin(t)cos(t)/d.
x² + y² = a²cos²(t) · (1 + sin²(t)) / d² = a²cos²(t) / d
(x² + y²)² = a&sup4;cos&sup4;(t) / d²
a²(x² − y²) = a&sup4;cos²(t)(1 − sin²(t)) / d² = a&sup4;cos&sup4;(t) / d² ✓
The denominator (1 + sin²(t)) is always in [1, 2] — never zero — so both lobes are traced continuously without any domain gaps.
Key Geometric Properties
| Property | Value (scale = a) |
|---|
| Width (tip to tip) | 2a |
| Max lobe height | ≈ 0.707 · a = a√2 / 2 |
| Crossing point | Origin (0, 0) |
| At rotation 0° | Lobes point left and right |
| At rotation 90° | Classic ∞ (lobes point up and down) |
Using the Controls
- Scale — the a parameter; half‑width of each lobe. Default 5 grid units.
- Rotation — tilts the entire curve. At 90° it looks like the ∞ symbol; at 0° the lobes point left and right.
- Pulse (accordion) — oscillates the scale sinusoidally, creating a breathing figure‑8. Speed (Hz) sets how many pulses per second; Amount sets how much the scale swells. The Scale slider updates live. Try combining with low Background opacity for glowing trails!
- Spin (accordion) — rotates the figure continuously about its center. Combine with Pulse for a mesmerizing animation.
- Background opacity — set low (5–25) to leave luminous motion trails each frame.
- Drag the center dot to reposition the figure anywhere on the canvas.
- Click the canvas to toggle the grid on/off.
Keyboard Shortcuts
- s — Toggle spin
- p — Toggle pulse
- r — Factory reset
- q — Stop all animations
📐 New to parametric equations? Take the guided tour →