A characterization of the sphere in terms of the stereographic projection

TL;DR

This paper characterizes the sphere in Euclidean space using properties of stereographic projection, focusing on symmetry and homothety conditions that distinguish the sphere from other convex bodies.

Contribution

It introduces two geometric conditions involving symmetry and homothety of sections that uniquely characterize the sphere via stereographic projection.

Findings

Spheres are characterized by axially symmetric cones from sections of the convex body.

The rotation invariance of cones relates to the homothety of sections under stereographic projection.

The stereographic projection maps circles onto circles, linking geometric properties to the sphere.

Abstract

Let $K$ be a convex body in the 3-dimensional Euclidian space $\mathbb{E}^3$ and let $N,S$ in the boubdary bd$K$ of $K$, $N\not=S$. Suppose that the support plane $\Pi_S$ of $K$ at $S$ is unique. For every point $x$ in bd$ K$, different than $N$, we define the stereographic projection $\Psi:\textrm{bd}K\backslash \{N\} \rightarrow \Pi_S$ of $x$ onto $\Pi_S$ as the point $y:=L(N,x)\cap \Pi_S$. It is a well known property of the sphere $\mathbb{S}^2$ in $\mathbb{E}^3$ that the stereographic projection maps circles onto circles (see \cite{Hilbert} pag. 248). In this work we investigate what geometric elements determines that this property is fulfilled. Here we demonstrate that the following two properties of a convex body $K\subset \mathbb{E}^3$ in terms of the stereographic projection characterize the sphere in $\mathbb{E}^3$: (1) The cones defined by the sections of $K$ and the point $N$ are axially symmetric (that is, they are invariant under a rotation by an angle of $\pi$). (2) given a section $K_\Gamma$ of $K$, the rotation that leaves the cone defined by $K_\Gamma$ and $N$ invariant is such that it maps $K_\Gamma$ into a homothetic figure to $\Psi(K_\Gamma)$ by a homothety with center of homothety at $N$. An important element in the proof of the main theorem of this work is a cha\-racterization of the circle based on a geometric property, which will be called the stereographic property. It is worth highlighting that the stereographic projection defined on the sphere maps circles onto circles is intimately linked to the conditions (1) and (2) and the stereographic property of the circle.