Sine–Cosine Track for a Swinging Spring Pendulum

This example visualizes a swinging spring pendulum as a dynamic, time-parameterized geometric construction, emphasizing the periodic nature of its motion and the resulting track/trajectory. It centers on sine and cosine relationships (amplitude and phase), coordinate geometry with parameterized points and Cartesian curves, and the way the path evolves as the time variable t changes. The construction uses GeoGebra commands such as Point (to define key moving and fixed points like D, P_1, P_2), If (to switch or clamp behavior under conditions), CurveCartesian (to generate the main trajectory curve in the coordinate plane, including forms like spiral-related paths), Segment (to draw connecting mechanical links), Sphere (for 3D geometric components), IntersectPath and Surface (to form intersections/trimmed geometric sets related to the moving configuration), plus Text objects for labeling parameters. The resulting interactive model is suitable for high school and early university learners studying trigonometry, analytic/coordinate geometry, and visual math modeling of periodic motion, especially in math–physics crossovers.