Geometric Visualization of the Torus–Sphere Relation Theorem

This example visualizes a torus–sphere relation theorem by constructing a sphere together with a parameter-defined surface in 3D and computing their intersections. It demonstrates how a geometric correspondence emerges from the way the two surfaces meet, highlighting symmetry and congruence-like relationships in space. The key concepts include 3D surfaces (a Sphere and a parametric Surface), intersection geometry (using intersection points that represent the shared locus), and the role of angular parameters (theta and phi) in describing the configuration. The construction uses GeoGebra commands such as Intersect (to find the intersection), Point (to place and track points), Surface and Sphere (to create the two surfaces), and Segment (to highlight and connect key elements). It also introduces parameter/angle objects (e.g., R, theta1, phi1, theta, phi, a, b) and annotation objects (text1–text4) to guide interpretation. Suitable for advanced secondary students and early university learners studying solid geometry, analytic/parameter geometry, and spatial intersection behavior.