On the constructibility of the axes of an ellipsoid

TL;DR

This paper examines Chasles's geometric construction for ellipsoid axes, identifying conditions under which the construction can be performed with classical tools and when it cannot, highlighting the geometric limitations.

Contribution

It proves the existence of non-planar cases in Chasles's construction and characterizes when the construction is feasible with compass and ruler.

Findings

Certain ellipsoid axes constructions are non-planar and cannot be done with classical tools.

The paper identifies specific conditions for planar and non-planar constructions.

It extends understanding of geometric constructibility related to ellipsoids.

Abstract

In this paper we discuss Chasles's construction on ellipsoid to draw the semi-axes from a complete system of conjugate diameters. We prove that there is such situation when the construction is not planar (the needed points cannot be constructed with compasses and ruler) and give some others in which the construction is planar.In this paper we discuss Chasles's construction on ellipsoid to draw the semi-axes from a complete system of conjugate diameters. We prove that there is such situation when the construction is not planar (the needed points cannot be constructed with compasses and ruler) and give some others in which the construction is planar.