Spherical Coordinates: Interactive 3D Visualization of Points on a Sphere

① This GeoGebra example dynamically demonstrates how a point in 3D space is determined by spherical coordinates (radius r, polar angle θ, and azimuthal angle φ), showing the point's position on a sphere as parameters change. ② Core mathematical concepts include conversion between spherical and Cartesian coordinates, sphere equations, parametric surfaces, and spatial point positioning, involving trigonometric functions and solid geometry. ③ Construction highlights: The model uses the GeoGebra command IntersectPath to generate intersection curves between the sphere and planes, with sliders controlling θ, φ, and r, and objects like p, q, d, e to build an interactive geometric setup that visualizes point movement and intersection paths. ④ Suitable for upper high school or early university students learning solid geometry, multivariable calculus, or physics applications of spherical coordinates, and for teachers to demonstrate coordinate transformations in class, enhancing students' 3D spatial reasoning.