Where is the cone?

TL;DR

This paper revisits the classical analytic relationship between conic sections and circular cones, providing a clear, modern linear algebra proof to fill a notable gap in mathematical literature.

Contribution

It offers a comprehensive and accessible analytic proof of the equivalence between conic sections and plane sections of cones, using standard linear algebra techniques.

Findings

Provides a complete analytic proof of the conic-cone equivalence.

Fills a historical gap in the mathematical literature.

Simplifies the understanding of conic sections through linear algebra.

Abstract

Real quadric curves are often referred to as "conic sections," implying that they can be realized as plane sections of circular cones. However, it seems that the details of this equivalence have been partially forgotten by the mathematical community. The definitive analytic treatment was given by Otto Staude in the 1880s and a non-technical description was given in the first chapter of Hilbert and Cohn-Vossen's "Geometry and the Imagination" (1932). The main theorem is elegant and easy to state but is surprisingly difficult to find in the literature. A synthetic version appears in The Universe of Conics (2016) but we still have not found a full analytic treatment written down. The goal of this note is to fill a surprising gap in the literature by advertising this beautiful theorem, and to provide the slickest possible analytic proof by using standard linear algebra that was not standard in 1932.