Non-Euclidean Triangle Centers

TL;DR

This paper explores the concept of triangle centers in non-Euclidean geometry, defining their homogeneous coordinates and extending classical notions like the median point and center of rotation to curved spaces.

Contribution

It introduces a unified coordinate framework for non-Euclidean triangle centers and generalizes key geometric points to curved spaces.

Findings

Homogeneous coordinates proportional to generalized sines are used for all spaces of uniform Gaussian curvature.

Definitions of median point and planar center of rotation are extended to non-Euclidean spaces.

The approach unifies Euclidean and non-Euclidean triangle center concepts.

Abstract

Non-Euclidean triangle centers can be described using homogeneous coordinates that are proportional to the generalized sines of the directed distances of a given center from the edges of the reference triangle. Identical homogeneous coordinates of a specific triangle center may be used for all spaces of uniform Gaussian curvature. We also define the median point for a set of points in non-Euclidean space and a planar center of rotation for a set of points in a non-Euclidean plane.