Stereographic Projection: Mapping a Sphere to a Plane

This example demonstrates the stereographic projection, a classic geometric transformation that maps points on a 3D sphere (except the projection pole) to a 2D plane via rays emanating from the pole, visually showing the conformal correspondence between the sphere and the plane. Core mathematical concepts include spherical geometry, projection transformations, intersections of lines with a sphere, and the images of parametric curves (e.g., latitude/longitude lines or arbitrary curves) under projection. Construction highlights: using GeoGebra's Sphere command to create the sphere, CurveCartesian to define parametric curves on the sphere, Point, Segment, and Vector to construct projection rays and directions, Mirror, Translate, and Polygon for symmetry and auxiliary shapes, Sequence to generate point sets or curve families, and Textfield and Slider (e.g., Taille, n, p) for interactive control. Suitable for high school to university-level teaching in solid geometry, analytic geometry, or topology, and for mathematical visualization to help students understand the mapping properties and conformality between sphere and plane.