Projections from surfaces of revolution in the Euclidean plane

TL;DR

This paper proves the existence of a smooth distance-preserving map from certain surfaces of revolution to the Euclidean plane, which simplifies their geometric structure by straightening meridians and parallels.

Contribution

It establishes a new class of maps that preserve infinitesimal distances along key curves of surfaces of revolution, linking these surfaces to the Euclidean plane.

Findings

Existence of a smooth map preserving distances along meridians and parallels.

The map sends meridional arcs to straight lines in the plane.

The result applies to a specific class of surfaces of revolution.

Abstract

For a specific class of surfaces of revolution S, the existence of a smooth map {\Phi} from a neighbourhood U of S to the Euclidean plane E2 preserving distances infinitesimally along the meridians and the parallels of S and sending the meridional arcs of U \ S to straight lines of E2; is proven.