M\"obius inversion of surfaces in the Minkowski 3-space

TL;DR

This paper explores the properties of M"obius inversion applied to surfaces in Minkowski 3-space, showing how it affects curvature lines, degenerate points, and ovaloids, with implications for geometric transformations.

Contribution

It introduces the concept of M"obius inversion in Minkowski space and analyzes its effects on surface properties, including curvature and ovaloid preservation.

Findings

M"obius inversion preserves principal curvature lines.

It maintains the locus of degenerate metric points.

It does not preserve the parabolic set.

Abstract

We define and present some proprieties of the M\"obius inversion of surfaces in the Minkowski 3-space. We prove that the M\"obius inversion preserves the lines of principal curvature and the locus of points where the metric is degenerate, but it does not preserve the parabolic set. For ovaloids, we show that it is possible to translate the surface so that the inversion remains an ovaloid.