Central Pole-to-Plane Projections on Ellipsoids and Elliptic Paraboloids: Geometry and Applications

In this work, we present an adaptation of the pole-based central projection from the sphere to the ellipsoid and the elliptic paraboloid. We begin by constructing the central pole-to-plane projections for each quadric surface separately, analyzing their geometric particularities and the challenges arising from variable curvatures and, in the case of the paraboloid, non-compactness. A key geometric insight reveals that the projected ellipses on the $xy$-plane and the corresponding conic sections on the quadrics are related by a homothety. This fundamental relationship allows us to establish unified scaling laws for their geometric invariants: the curvature scales by $\lambda^{-1}$, the arc length by $\lambda$, and the area by $\lambda^2$, where $\lambda$ is the homothety factor. These results provide a complete characterization of the eccentricities, curvatures, arc lengths, and areas of the intersecting conics and their projections.