A1 Refereed original research article in a scientific journal
Collinearity of points on Poincaré unit disk and Riemann sphere
Authors: Fujimura, Masayo; Rainio, Oona; Vuorinen, Matti
Publisher: Institute of Mathematics, University of Debrecen
Publication year: 2024
Journal: Publicationes Mathematicae Debrecen
Journal name in source: Publicationes Mathematicae Debrecen
Volume: 105
Issue: 1-2
First page : 141
Last page: 169
ISSN: 0033-3883
eISSN: 2064 - 2849
DOI: https://doi.org/10.5486/PMD.2024.9763(external)
Web address : https://doi.org/10.5486/PMD.2024.9763(external)
Self-archived copy’s web address: https://arxiv.org/abs/2212.09037(external)
Preprint address: https://arxiv.org/abs/2212.09037v1(external)
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 Gröbner bases from computer algebra for the solution of polynomial equations.
Funding information in the publication:
The first author was partially supported by JSPS KAKENHI (Grant No. JP19K03531), and the second author was supported by Finnish Culture Foundation and Magnus Ehrnrooth Foundation.
