What do you see when two oscillations collide at right angles? Tune the frequency ratio, phase and damping to explore the mathematical beauty found on every oscilloscope.
Oscillation Parameters
Frequency f_x
Frequency f_y
Phase difference φ
Amplitude Ax
Amplitude Ay
Drawing settings
Number of points
Line width
Color mode
Animation
Harmonograph
Damping d_x
Damping d_y
Presets
a:b = 3:2φ = 0.00πButterfly
Live quantities
3:2
Frequency ratio a:b
3
Frequency a
2
Frequency b
0°
Phase δ
1.00 / 1.00
Amplitude A / B
Yes
Closed curve?
2 / 3
Lobes x / y
1.00
Period [×2π]
Oscilloscope (XY mode) — two oscillations draw the figure
Phase presets:
x oscillation (top edge)y oscillation (left edge)Tracer point (x,y)
Check against known patterns
1:1, δ=0° → diagonal line (preset “1:1 45°” with δ set to 0)
1:1, δ=90° → circle / ellipse
1:2, δ=0° → figure eight (2 lobes)
2:3 → butterfly (3 lobes in x, 2 lobes in y)
f_x : f_y = (number of touches on the left/right edges) : (number of touches on the top/bottom edges). With an irrational ratio (e.g. 1:√2) the figure never closes. The sliders move in 0.01 steps, so with a non-integer value (e.g. 3.07) the ratio, lobe count and period are shown as "—".
Theory & Key Formulas
$x(t) = A \sin(a\,t + \delta)$
$y(t) = B \sin(b\,t)$
If the frequency ratio $a:b$ is rational the curve is closed. The phase $\delta$ sets its orientation.
Harmonograph (damped):
$x = A\,e^{-d_x t}\sin(a\,t+\delta)$
What Are Lissajous Figures?
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What exactly is a Lissajous figure? In the simulator it looks like a point is moving, but why does it draw such a beautiful pattern?
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Basically, it is the path traced by the tip when two independent oscillations at right angles to each other are combined. Imagine swinging a pendulum north–south while also pushing it east–west. The position of the point at any moment combines the X coordinate and the Y coordinate from each motion. Set both frequencies (f_x and f_y) to 1 in the simulator. You should see a simple ellipse or circle.
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Really!? So the shape of the pattern is decided by the two frequencies. What happens if I make them different, like 1 and 2?
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Exactly! The frequency ratio is the key. A ratio of 1:2 means that the X direction completes only one cycle while the Y direction completes two. This produces the classic "figure-eight" pattern. Actually set f_x=1 and f_y=2 and move the "Phase difference φ" slider little by little. You can watch the figure-eight rotate and deform!
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That's fascinating. What does the "damping" slider do? When I increase it, the curve spirals inwards.
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Good observation! Damping simulates real-world energy loss such as friction. It makes the amplitude of each oscillation decay exponentially with time. So instead of a perfectly closed ellipse repeating forever, the curve spirals towards the centre as the amplitude shrinks. In automotive suspension testing, for example, engineers add damping and check how the oscillation settles. Turn the harmonograph ON and set damping d_x=50 and d_y=20 and an asymmetric spiral appears.
Physical Model and Key Formulas
Inside the simulator are two independent damped harmonic oscillators. The horizontal (X) and vertical (Y) positions of the tracing point are given by the following equations, which combine a sine wave with an exponential decay term.
Meaning of the variables:
$A_x, A_y$: initial amplitudes (size of the swing).
$f_x, f_y$: relative frequencies (1.00–8.00; they have no unit, and only their ratio decides the figure).
$d_x, d_y$: damping coefficients (harmonograph mode only; slider values 0–50 correspond to internal coefficients 0–0.005).
$\varphi$: phase difference (the time offset between the X and Y oscillations).
$t$: the parameter along the curve (dimensionless; without damping one period until the figure closes is drawn, and in the harmonograph $t=0$ to $40\pi$).
The resulting figure is a parametric curve plotting $y(t)$ against $x(t)$. Only when the frequency ratio $f_x : f_y$ is rational (e.g. 3:2) and the damping is zero does the figure close and repeat periodically. The phase difference $\varphi$ controls the "rotation" or orientation of the pattern.
$$ \frac{f_x}{f_y}= \frac{m}{n}\quad \text{(condition for a stable closed curve)} $$
Here $m$ and $n$ are integers. A 1:1 ratio ($m=n$) produces an ellipse, and a 2:3 ratio a more complex looped figure. With damping, the figure does not close and spirals inwards.
Real-World Applications
Analogue oscilloscope calibration:Before digital oscilloscopes became common, engineers used Lissajous figures to measure unknown frequencies. A known frequency is fed to one channel and the unknown signal to the other, and the frequency ratio is read from the shape of the figure that appears. A stable circle means the two frequencies match exactly.
Mechanical vibration analysis:In rotating machinery, displacement sensors in two perpendicular directions are placed near the bearings, and the path of the shaft centre (the orbit) is drawn on an X-Y display. The shape of the orbit gives clues for distinguishing faults such as unbalance (an ellipse) and misalignment (a shape close to a figure-eight).
Laser light shows:The fantastic patterns of laser displays are often made by precisely oscillating mirrors arranged at right angles. By adjusting the mirrors' frequencies and phases (in the same way as this simulator's sliders), all kinds of patterns can be produced, from simple spirals to complex butterfly-like figures.
Phase comparison of electrical signals:In AC circuit analysis, comparing the phase difference between voltage and current is extremely important. Displaying both on an X-Y plot draws a Lissajous ellipse. The phase difference can be read directly from the tilt and shape of the ellipse, which helps diagnose power-factor problems.
Common Misconceptions and Points to Note
First,don't assume the frequency ratio is limited to simple integer ratios. Ratios such as 1:2 or 3:4 certainly produce beautiful closed figures, but with an irrational ratio such as 1:√2 the figure never closes and gradually fills the whole drawing area over time. This is called "quasi-periodic motion" and is important in analysing anharmonic vibrations in the real world. Next,the interpretation of the "phase difference" also needs care. Some literature adds the phase to the y oscillation ($y = B\sin(b\,t + \delta)$), but this tool adds φ to the x oscillation (x leads y by φ). Reading literature without being aware of this difference in definition causes confusion, so in practice always check which definition is used.
As a tip for setting parameters, pay attention tothe balance between the "number of points" and the size of the frequencies. This tool draws one period until the figure closes, divided into the "number of points". For ratios with larger frequencies such as 5:4 or 7:8, one period contains more oscillations, so with few points the curve looks angular. In that case increase the number of points. The harmonograph draws $t=0$ to $40\pi$, and even at maximum damping (internal coefficient 0.005) the amplitude at the end is about 0.53 times, so the figure never shrinks to a single point at the centre. Finally,the amplitudes Ax and Ay affect not only the "apparent size" of the figure but also its "shape". In particular, when the damping coefficients dx and dy are asymmetric, differences in amplitude have a dramatic effect on how the spiral is drawn in. Try adjusting them together.
Set the relative frequency f_x in the X direction in the range 1.00–8.00 (slider values 100–800, no unit)
Set the relative frequency f_y in the Y direction in the same range. The figure is decided by the ratio f_x : f_y; with integer values, the ratio, lobe count and whether the curve is closed are shown
Adjust the phase difference φ from 0 to 2π rad (slider values 0–628, equivalent to 0–360°) to rotate or flip the Lissajous curve. Shifting the phase produces asymmetric figures
Change the amplitudes Ax and Ay from 0.1 to 1.0 (slider values 10–100) to stretch the pattern vertically or horizontally
To imitate the energy loss of mechanical vibration, turn on harmonograph mode and set the damping d_x and d_y (slider values 0–50, internal coefficients 0–0.005)
The curve is redrawn as soon as you move a slider. "Save PNG" saves it as an image
Worked Example
Set f_x=3, f_y=2, phase=45°, amplitude=1.0 and damping=0. With the 3:2 frequency ratio, a "butterfly" Lissajous curve is drawn that touches the left/right edges three times and the top/bottom edges twice, with two lobes in the x direction and three in the y direction (readout "Lobes x / y" = 2 / 3). Changing the phase to 90° (slider value 157) keeps the curve closed and only changes its shape (a tilted ellipse appears only for a 1:1 ratio). Turning the harmonograph ON with damping d_x=d_y=50 (internal coefficient 0.005), the amplitude at the end of the drawn interval (t=40π) has decayed to exp(−0.005×40π) ≈ 0.53 times, and the curve spirals gently inwards from the outer trace. This is a simplified imitation of the damped behaviour of harmonic oscillation in real mechanical systems such as seismic isolation dampers and vibration control devices.
Practical Notes
Integer ratios (e.g. 1:1, 2:3, 5:4) give closed curves, and the frequency ratio can be read from the number of edge touches. This is the principle used when comparing the frequencies of two signals on an oscilloscope
The less simple the ratio (e.g. 7:5), the more back-and-forth motion is needed before the curve closes, and the figure becomes denser. With a strictly irrational ratio it never closes and fills the area quasi-periodically
Adjusting the phase offset can be applied to simulating phase lag in AC electrical measurement (e.g. power-factor analysis at 50–60 Hz) or timing offsets in dual-motor drive systems
The damping d_x and d_y in this tool is an envelope decay for drawing (internal coefficient up to 0.005) and is defined differently from the damping ratio ζ of real machines. For evaluating real systems, carry out a separate vibration analysis using the damping ratio ζ