Lissajous Curve Visualizer
Interactive visualization of Lissajous curves — the parametric curves traced by two perpendicular simple harmonic motions. Adjust the frequency ratio, phase difference, and amplitudes to explore how these elegant patterns form, decompose, and fill 3D space.
2D Lissajous Curve
The curve traces as t advances. The faint background shows the complete figure; the bright trail is the live trace.
3D Representation
The curve plotted in (x, y, t) space. Drag to rotate. The dashed shadow is the 2D projection.
x(t) = A·sin(at + δ) — Horizontal Component
y(t) = B·sin(bt) — Vertical Component
Controls
About Lissajous Curves
A Lissajous curve is the trajectory of a point undergoing two perpendicular simple harmonic motions. The horizontal position follows and the vertical position follows . The shape depends critically on the frequency ratio a:b and the phase difference δ.
- Closed curves form when a/b is rational. The curve closes after period T = 2π / GCD(a, b).
- Phase δ = π/2 with equal frequencies produces a circle (or ellipse, if amplitudes differ).
- δ = 0 produces a straight line when a = b, or a Lissajous figure that touches the bounding box edges.
- 3D view shows how x and y oscillate independently as t advances — the 2D curve is the projection onto the xy-plane.