Central cross-sections make surfaces of revolution quadric

TL;DR

This paper proves that surfaces of revolution with centrally symmetric cross-sections are necessarily quadric surfaces, establishing a unique geometric characterization of quadrics among such surfaces.

Contribution

It introduces a new geometric criterion based on central symmetry of cross-sections to identify quadrics among surfaces of revolution.

Findings

Surfaces of revolution with symmetric cross-sections are necessarily quadrics.

Quadrics are the only revolution surfaces without skewloops.

The result extends to higher-dimensional hypersurfaces.

Abstract

We prove here that when all planes transverse and nearly perpendicular to the axis of a surface of revolution intersect it in loops having central symmetry, the surface must be quadric. It follows that the quadrics are the only surfaces of revolution without skewloops. Similar statements hold for hypersurfaces of revolution in higher dimensions.