Lissajous Figures

Lissajous curves are the graphs of parametric equations that describe complex harmonic motion. They appear whenever two perpendicular oscillations are combined — for instance on an oscilloscope when two sinusoidal signals are fed into the X and Y channels.

The Equations

A point traces out a Lissajous figure when its $(x, y)$ coordinates are driven by two independent sinusoidal functions:

$x(t) = A\sin(a\,t + \delta) \qquad y(t) = B\sin(b\,t)$

Parameter	Meaning
$A$, $B$	Amplitudes — control the size of the figure in x and y
$a$, $b$	Frequencies — their ratio $a : b$ determines the shape's complexity
$\delta$	Phase shift — rotates / morphs the figure continuously

Why the Ratio Matters

• When $a : b$ is a simple ratio (like 1:1, 1:2, 3:2) the curve closes and repeats after a finite time.
• When $a : b$ is irrational (e.g. $\sqrt{2} : 1$) the curve never closes — it fills a region of space over infinite time.
• The number of lobes / crossings increases with the complexity of the ratio.

Phase Shift

At $\delta = 0$ or $\delta = \pi$ with $a = b$, you get a straight line (the two oscillations are perfectly in-phase or anti-phase). At $\delta = \pi/2$ with $a = b$, you get a circle or ellipse.

---

---

Things to Try

1. Circle / Ellipse: Set $a = b = 1$, $\delta = \pi/2$ (1.57). You'll see a perfect ellipse. Make $A = B$ and it becomes a circle.
2. Figure-8: Set $a = 1$, $b = 2$, $\delta \approx \pi/2$.
3. Infinity symbol: $a = 2$, $b = 1$ traces the ∞ shape.
4. Slowly sweep δ: Keep $a = 3$, $b = 2$ and drag δ from 0 to 2π. Watch the figure morph through strikingly different patterns.
5. High complexity: Try $a = 7$, $b = 5$ — the number of lobes multiplies with the ratio's numerator and denominator.

Where Lissajous Figures Appear

• Oscilloscopes: Historically used to compare two signal frequencies. The shape tells you the frequency ratio at a glance.
• Music & Acoustics: Tuning forks held at right angles trace these patterns on smoked glass (a 19th-century experiment).
• Laser shows: Mirrors vibrating at precise frequencies draw Lissajous patterns with laser beams.
• Mechanical engineering: Vibration analysis of coupled oscillators reveals Lissajous-like motion in structural modes.

---

Now Add a Third Dimension — and Watch Them Come Alive

What happens when you add a third oscillation perpendicular to the first two? The flat curves leap off the page into sculpted 3D forms — knots, helices, and impossible ribbons that twist through space. Drag to rotate, scroll to zoom, and sweep the parameters to discover shapes you've never seen before.

$x(t) = A\sin(a\,t + \delta_{xy}) \qquad y(t) = B\sin(b\,t) \qquad z(t) = C\sin(c\,t + \delta_{xz})$