The Integral of Secant and Stereographic Projections of Conic Sections

TL;DR

This paper explores how stereographic projections of conic sections underpin the four main substitutions used to integrate secant, revealing their geometric origins and connections.

Contribution

It uncovers the geometric basis of secant integration substitutions through stereographic projections and analyzes their interrelations.

Findings

The four secant integration substitutions are based on different rational parametrizations.

Connections between these substitutions are identified and explained.

Stereographic projections provide a unified geometric framework for understanding these methods.

Abstract

We show that the four best-known substitutions used to integrate secant rely on different rational parametrizations of conic sections, coming from stereographic projections. We also investigate other possible explanations for the four substitutions, as well as connections between the buildups leading to them.