Generate Lissajous figures by adjusting frequency ratios, phase differences, and damping. Animate the drawing process.
A Lissajous figure is the path formed when one sinusoidal motion supplies the horizontal coordinate and another supplies the vertical coordinate. This generator lets you vary their frequency ratio, shift their relative phase, add damping, and animate the order in which the path is drawn. The common surprise is that the frequency numbers do not uniquely determine the completed undamped shape. Ratios 2:3 and 4:6 reduce to the same ratio, so over a complete repeat they trace the same locus, although their traversal timing can differ.
A concrete input and expected output from the current implementation.
Input
frequency ratio 1:1; phase difference 0 degrees; damping 0
Expected output
A straight line segment through the center, traced back and forth without contracting.
With equal frequencies and zero phase difference, the normalized coordinates have the same sine value at every parameter time: x(t) = sin(t) and y(t) = sin(t), so y = x. Zero damping keeps the amplitude constant.
For interpreting an undamped plot, write the coordinates as x(t) = A sin(a t + delta) and y(t) = B sin(b t). The values a and b set the frequency ratio, while delta offsets one oscillation in phase. If a:b is rational, the two motions regain the same relative state after a common period and the full trace closes.
Positive damping makes the oscillation amplitude decrease as parameter time advances. Consequently, a cycle drawn later is smaller than an earlier cycle, and repeated lobes no longer lie exactly on top of one another. The animation reveals which portion is earlier and which contracts later, information that is ambiguous in a finished still image.
No. With damping set to zero, a Lissajous trace closes whenever the frequency ratio is rational, such as 2:3, because both oscillations return to their starting relationship after a common period. Equal frequencies are only the 1:1 special case. A short animation can stop before closure, so an apparently open path does not disprove eventual closure.
Phase sets the relative starting point of the two oscillations. At 1:1, zero phase makes their normalized coordinates equal at every instant, producing a line. A quarter-cycle offset puts them in quadrature; with equal amplitudes and equal axis scales, the mathematical locus is a circle. Other nondegenerate offsets produce ellipses. Positive damping prevents later cycles from exactly retracing that locus.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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