Mathematics for dancing laser beam

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Discussion Overview

The discussion revolves around the mathematical description of shapes created by laser beams, particularly in relation to Lissajous curves and their connection to music frequencies. Participants explore the theoretical and practical aspects of these shapes, including their generation and characteristics in the context of chaos theory and differential equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that the shapes can be described by chaos theory, specifically through strange attractors and closed paths in vector fields of ordinary differential equations.
  • One participant shares a personal experience of creating Lissajous figures using a 2-D laser deflection mirror driven by music signals, highlighting the fun and artistic aspect of the project.
  • Another participant notes that while Lissajous curves are common, there are instances where closed curves appear that do not resemble these figures.
  • There is a suggestion that the observed patterns may indicate two waves that are out of phase, with one participant describing the wave shapes as potentially non-sinusoidal.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the shapes produced, with some supporting the idea of Lissajous curves while others point out the existence of different closed curves. The discussion remains unresolved regarding the definitive mathematical characterization of these shapes.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the waveforms and their relationships, as well as the specific mathematical frameworks that may apply to the shapes being discussed.

sergiokapone
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Is there a math that describes these shapes at least one frequency?
 
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I would say that these are strange attractors that are dealt with in chaos theory. The more regular shapes are closed paths in the vector fields of ordinary differential equations, basically also attractors.
 
sergiokapone said:
Is there a math that describes these shapes at least one frequency?
Back in undergrad, I made a 2-D laser deflection mirror that I could drive in the horizontal axis with a lower-frequency waveform (like 50Hz-100Hz), and I drove the music signal into the vertical deflection circuit. You could adjust the horizontal sinusoid to get the best Lissajous figures for each particular piece of (rock) music. Very fun...

https://en.wikipedia.org/wiki/Lissajous_curve
 
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Well, if there were always Lissajous curves, then everything is fine. But sometimes closed curves are obtained, which are far from similar to such figures.
 
sergiokapone said:
Well, if there were always Lissajous curves, then everything is fine. But sometimes closed curves are obtained, which are far from similar to such figures.
Can you e-mail the person who created that figure and other similar figures? Perhaps they would share their setup details with you.
 
It looks as if there are just two cycles of X and Y, as I can see ends to the pattern. The existence of an enclosed area means the two waves are out of phase, having a quadrature component. It looks as if they are the same frequency, as we do not see a figure-of-eight in the pattern, but the wave shapes look non sinusoidal, maybe rectified sine wave.
 
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berkeman said:
Back in undergrad, I made a 2-D laser deflection mirror that I could drive in the horizontal axis with a lower-frequency waveform (like 50Hz-100Hz), and I drove the music signal into the vertical deflection circuit. You could adjust the horizontal sinusoid to get the best Lissajous figures for each particular piece of (rock) music. Very fun...

BTW, this was one of the better musical pieces for the laser show in our dorm room, and this thread reminded me of it so I had to post it. It's also the song that I get in my head when I'm pushing the cadence of my mountain bike (MTB) to keep a high pedal cadence. :smile:

Set your Lissajous laser deflection at 60Hz horizontal and turn up the vertical and the audio!


1663288859087.png
 
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