Analytic sphere eversion using ruled surfaces

TL;DR

This paper presents explicit formulas for sphere eversion using ruled surfaces, simplifying the process both analytically and topologically, and demonstrates their utility in visualization with standard software.

Contribution

It introduces a new analytic approach to sphere eversion via ruled surfaces, making the process more accessible and easier to visualize.

Findings

Explicit formulas for sphere eversion are derived.

The method includes the Boy surface as a special case.

Visualization of the eversion process is facilitated with common software.

Abstract

Sphere eversions have been described so far by either pictures with minimal topological complexity, numerical evolution or complex equations. We write down relatively simple explicit formulas for the whole eversion, both analytic and topologically simpler, including also Boy surface (real projective plane), using a family of ruled surfaces. We show their usefulness in visualizing the process using commonly available modeling software.