Lissajous Illustrator and Animator

Interactive tool for drawing and animating Lissajous curves. Adjustable frequencies, phase shifts, colors, and curve thicknesses for custom designs.
Lissajous curves or figures are curve graphs that result from the perpendicular superposition of two sine waves. See below for further mathematical and physical explanations.

The background and curve colors, as well as the curve thickness, can be freely chosen. Rainbow is a special visual effect using spectral colors that replace the curve color. These settings have no mathematical significance. The following settings are available for the calculation:
• The most important value for defining a Lissajous curve is the frequency ratio. The more dissimilar the two numbers in the ratio are, the more complex the Lissajous figure becomes. The default is 3:4. Interesting effects can be achieved with two- or three-digit coprime numbers.
• The phase shift indicates how far the two sine waves are offset from each other. This value is important for static images. For animations, it merely defines the starting point. Phase shifts can be entered as numbers or as calculations using basic arithmetic operations, with pi for the particularly important number π. Therefore, entries like pi/2+1 are also permitted.
• The amplitude ratio indicates the stretching left/right or up/down. Often, this remains at 1. In this case, a grid can also be drawn, with the spacing between each line being 0.1.
• The cycle factor specifies how many times the curve completes a full rotation. If nothing is specified, the value chosen is the one that draws the entire figure once. Small values ​​result in the curve being drawn incompletely. Large values ​​lead to unexpected graphic artefacts.
• Finally, the animation speed determines how fast the figure rotates. The value is between 0 and 500, where 0 means no rotation. Buttons for saving the animation as a video are available below the image. The direction of rotation is irrelevant. The rotation can be viewed clockwise or counterclockwise, as the apparent third dimension is purely an optical effect.

Settings Background color:
Curve color:
Curve thickness:
Image size:
Frequency ratio: :
Phase shift:

Amplitude ratio: :

Cycle factor:



* Animation speed:



* Warning! Animations may flicker and pose risks to sensitive individuals. This is especially true for people with photosensitive epilepsy. If you are prone to epileptic seizures or dizziness, do not start the animation.




Conventional saving, for example by right-clicking, only saves an image, not the animation. However, the buttons above allow you to record videos of the animation in WebM format. Start Recording begins the moment the button is pressed, Start Animation and Recording starts recording at the beginning of the animation. But first, the animation has to be started at the desired speed. Stop and Save Video downloads the recorded video. Depending on its length, this can be quite large.

A Lissajous figure with a frequency ratio of 3:4, as set here by default, is generated from two sine waves whose x-coordinate is described by x(t) = sin(3t) and whose y-coordinate by y(t) = sin(4t). For other frequency ratios, the values ​​are correspondingly different. t is a parameter often interpreted as time. The phase shift δ is added to one of the two sine functions, for example, x(t) = sin(3t + δ).

Lissajous curves were discovered in the mid-19th century by the French physicist Jules Antoine Lissajous and later named after him. They can be displayed on an oscilloscope, where they serve to illustrate alternating currents. Due to their aesthetic qualities, they are used as models for art and design.

The graphics and animations generated with this application are not protected by copyright and can therefore be used freely.


Rechneronline | Impressum & Datenschutz | German: Lissajous Figuren

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