Spiric Sections: Exploring Torus Cross-Sections via Parametric Surfaces

This GeoGebra example demonstrates Spiric sections, the classical plane curves obtained by intersecting a torus with a plane parallel to its axis. It visualizes how the cross-section evolves from circles and ellipses to lemniscate-like shapes as parameters change. Core mathematical concepts include parametric surfaces, parametric curves, torus geometry, and the relationship between 3D solids and their 2D sections. The construction uses the Surface command to render the torus and CurveCartesian to plot the section curve, with sliders controlling parameters a, n, b, c, and d for dynamic exploration. Suitable for high school and university students studying analytic geometry, parametric equations, and 3D visualization, this interactive model enhances understanding of how algebraic parameters shape geometric loci.