Curvature types of planar curves for gauges

TL;DR

This paper extends the differential geometry of curves from normed to gauge planes, classifying all curvature types and generalizing concepts like evolutes and involutes in this broader setting.

Contribution

It introduces a complete classification of curvature types in gauge planes, generalizing the theory from normed planes and extending key geometric notions.

Findings

Four curvature types identified: Minkowski, normal, circular, arc-length

Relations between curvature types established

Evolutes and involutes extended to gauge planes

Abstract

In this paper results from the differential geometry of curves are extended from normed planes to gauge planes which are obtained by neglecting the symmetry axiom. Based on the gauge analogue of the notion of Birkhoff orthogonality from Banach space theory, we study all curvature types of curves in gauge planes, thus generalizing their complete classification for normed planes. We show that (as in the subcase of normed planes) there are four such types, and we call them analogously Minkowski, normal, circular, and arc-length curvature. We study relations between them and extend, based on this, also the notions of evolutes and involutes to gauge planes.