Revisiting floating bodies

TL;DR

This paper provides new characterizations of conic sections and quadrics based on tangent line and plane segments of constant area or volume, extending classical results to higher dimensions.

Contribution

It introduces novel characterizations of conic sections and quadrics using tangent segments of constant measure, generalizing classical geometric properties to multidimensional cases.

Findings

Characterization of conics via tangent segments of constant area

Extension of properties to quadrics and higher dimensions

Generalization of classical geometric results

Abstract

The conic sections, as well as the solids obtained by revolving these curves, and many of their surprising properties, were already studied by Greek mathematicians since at least the fourth century B.C. Some of these properties come to the light, or are rediscovered, from time to time. In this paper we characterize the conic sections as the plane curves whose tangent lines cut off from a certain similar curve segments of constant area. We also characterize some quadrics as the surfaces whose tangent planes cut off from a certain similar surface compact sets of constant volume. Our work is developed in the most general multidimensional case.