Collinearity of points on Poincar\'e unit disk and Riemann sphere

TL;DR

This paper explores the collinearity of special points in hyperbolic geometry on the Poincaré disk and Riemann sphere, providing explicit formulas and new methods for midpoints using algebraic techniques.

Contribution

It introduces explicit formulas for intersection points and new collinearity results, employing computer algebra to solve polynomial equations in hyperbolic geometry.

Findings

Explicit formulas for intersection points on the Poincaré disk and Riemann sphere

New collinearity theorems for hyperbolic geometry points

Algebraic methods for finding hyperbolic and chordal midpoints

Abstract

We study certain points significant for the hyperbolic geometry of the unit disk. We give explicit formulas for the intersection points of the Euclidean lines and the stereographic projections of great circles of the Riemann sphere passing through these points. We prove several results related to collinearity of these intersection points, offer new ways to find the hyperbolic midpoint, and represent a formula for the chordal midpoint. The proofs utilize Gr\"obner bases from computer algebra for the solution of polynomial equations.