Stereographic Projections on Some Quadric Surfaces

TL;DR

This paper extends the classical stereographic projection to ellipsoids and elliptic paraboloids, analyzing geometric properties and intersection characteristics of these quadric surfaces.

Contribution

It introduces adapted stereographic projections for ellipsoids and paraboloids, addressing their unique geometric features and challenges.

Findings

Derived projections for ellipsoids and paraboloids.

Analyzed curvature, eccentricity, and arc length of intersections.

Compared geometric properties of projected quadrics.

Abstract

In this work, we present an adaptation of the classical stereographic projection, originally formulated for the sphere, now considering the context of the ellipsoid and the elliptic paraboloid. We begin by constructing the stereographic projections for both quadric surfaces separately, analyzing the geometric particularities of each surface and the challenges arising from their variable curvatures and, in the case of the paraboloid, its non-compactness. In the final part of the work, we establish results concerning the eccentricities, curvatures, arc length, and areas of the ellipses formed by the intersection of the quadrics with horizontal sections and their corresponding projections onto the plane-xy.