Viviani’s Window: Parametric Curves from a Cylinder–Sphere Intersection

This example demonstrates Viviani’s Window by visualizing how a cylinder and a sphere intersect to produce characteristic spatial curves and “window-like” contours. It highlights the geometric relationship of the intersection region, then connects that geometry to the idea of integral calculus (e.g., using integration to relate accumulated quantities such as area/volume). Key concepts include solid geometry (cylinder/sphere intersection), parametric curve modeling, and calculus workflows that combine arithmetic operations with function evaluations such as logarithmic expressions. In GeoGebra, the construction centers on CurveCartesian to create the main curve geometry (objects named testo1, testo2, testo3, testo4). Parameters R, b, c, and d define the geometry, while the variable speed drives or controls the dynamic aspect of the visualization. Suitable for high school to undergraduate students studying solid geometry, coordinate/parametric curve representations, and introductory calculus using interactive visualization.