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Harmonograph: Parametric Curves from Swinging Pendulums

Harmonograph: Parametric Curves from Swinging Pendulums
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2023-03-28 03:20
TrigonometryParametric EquationSlider AnimationLocusGraphing
In collections Música y Matemáticas
This GeoGebra demonstration illustrates the mathematical principles of a harmonograph, a mechanical device that creates intricate curve patterns through two swinging pendulums. The core mathematical concepts involve parametric equations, trigonometric functions, and locus generation. Construction details: Using the Segment command to create pendulum arms, with parameters a and b controlling the lengths of two pendulums, 'desfase' (phase shift) adjusting the time offset between swings, and 'tiempo' (time) controlling the animation progression to generate periodic decaying complex curves. As shown in the construction summary, multiple variables s1-s4 and position points A, Ax, Ay, B1-B7 work together to produce the final trajectory. Suitable for upper secondary and university mathematics classes, demonstrating applications of trigonometric functions in physical motion, dynamic visualization of parametric equations, and the interdisciplinary connection between mathematics and music.

GeoGebra Commands & Parameters

  1. Segment(Ay, A) → s1
  2. Segment(A, Ax) → s2
  3. Segment((-1, nada), (-1 + nada, 10)) → s3
  4. Segment((nada, -1), (10, -1 + nada)) → s4

AI Drawing Prompt Reference

Create a harmonograph-style parametric curve on a coordinate plane. Use two swinging-pendulum motions controlled by length parameters a and b, a phase shift (desfase) to offset the timing between the two swings, and a time parameter (tiempo) to advance the animation and produce periodic behavior with a decaying, intricate trajectory. Construct the setup from a reference pivot point, then form two perpendicular baseline segments along the horizontal and vertical directions whose endpoints determine the projection geometry for the two pendulum arms. Combine the corresponding trigonometric time-dependent components into parametric displacement functions and render the resulting locus/trace as the final curve.
The prompt above was compiled from the original drawing commands and is for reference only.