Conic Section as a Cone–Plane Intersection in Cylindrical Coordinates

This example demonstrates how to visualize a conic section (e.g., an ellipse) as the intersection curve between a cone surface and a plane in 3D space. It focuses on the key idea that conic sections arise from slicing quadratic surfaces, and it uses cylindrical-coordinate parameterization to describe the geometry. The construction uses GeoGebra commands such as Surface and CurveCartesian to build the relevant surfaces and the resulting Cartesian parameter curve. Internally, the model defines parameter mappings and intersection-related quantities (e.g., h_r, r, r_x, K_r, Z_r) to convert cylindrical coordinates into 3D coordinates, then introduces parameters like q and labeling objects (text1, text3). Additional variables (fx, fy, fz, E, a, K_{KxE}, EP) help assemble the cone/plane intersection constraints into computable parametric expressions, yielding the final intersection locus. Suitable for high school to early undergraduate learners studying 3D analytic geometry, parametric equations, and conic sections.