3D Oscillating Pendulum and Lissajous Figures Animation

This GeoGebra example demonstrates an oscillating pendulum in three-dimensional space and generates Lissajous figures through animation. Lissajous curves are formed by the superposition of two perpendicular sinusoidal vibrations, displaying classic patterns that reveal the relationship between phase difference, frequency ratio, and amplitude. The core mathematical concepts involved include trigonometric functions (sine/cosine), parametric equations, and the principle of vibration superposition. Construction details: The CurveCartesian command is used to plot parametric curves, while the Segment command constructs the pendulum linkage. The parameter t serves as the time variable controlling the x-direction vibration f_x = sin(f_1·t + φ) and y-direction vibration f_y = sin(f_2·t). Objects such as φ (phase difference), A_x, A_y (amplitudes), and f_1, f_2 (frequency ratio) enable dynamic animation effects, with Slider controls for parameter adjustment and observation of how Lissajous patterns transform. This example is ideal for high school mathematics (trigonometry and parametric equations) and introductory university physics (vibrations and waves) instruction, serving as an excellent tool for visualizing the beauty of mathematics and its interdisciplinary applications.