On the shadow problem and its generalizations to ellipsoids

TL;DR

This paper extends the classical shadow problem from spheres to ellipsoids, aiming to determine the minimal configuration of non-overlapping balls that intersect all lines through the ellipsoid's center, and proposes a new solution method.

Contribution

It generalizes the shadow problem to ellipsoids and introduces a novel approach for solving it, expanding understanding beyond the sphere case.

Findings

Determined minimal configurations for the shadow problem on ellipsoids

Proposed a new method for solving the shadow problem

Extended classical results from spheres to ellipsoids

Abstract

The main goal of the paper is to solve some problems about shadow for the sphere generalized on the case of the ellipsoid. Here, the essence of the problem is to find the the minimal number of non-overlapping balls with centers on the sphere which are not holding the center of the sphere and such that every line passing through the center of the sphere would intersect at least one of the balls. Another method of solving the shadow problem for the sphere is also proposed.