Thread configurations for ellipsoids

TL;DR

This paper explores special thread configurations on ellipsoids involving segments and geodesics that maintain constant length when vertices move along confocal ellipsoids, extending classical configurations to three dimensions.

Contribution

It introduces Darboux-Staude type thread configurations for ellipsoids, generalizing known ellipse configurations to three-dimensional ellipsoids with fixed-length properties.

Findings

Defined new thread configurations on ellipsoids involving geodesic and line segments.

Proved these configurations preserve length when vertices move on confocal ellipsoids.

Extended classical ellipse thread configurations to three-dimensional ellipsoids.

Abstract

We discuss Darboux-Staude type of thread configurations for the ellipsoid similar to Chasles-Graves type of thread configurations for the ellipse. These threads are formed by rectilinear segments, geodesic and line of curvature segments on the considered ellipsoid and with tangents tangent to the given ellipsoid and a fixed confocal hyperboloid with one sheet and preserve constant length when the vertices of the configuration move on confocal ellipsoids.