Non-Euclidean Triangle Centers

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.

Figures (12)

• Figure 1: Regardless of the Gaussian curvature $K$, the homogeneous coordinates of the point $M$ are $(\mathop{\rm sing}\nolimits h_a:\mathop{\rm sing}\nolimits h_b:\mathop{\rm sing}\nolimits h_c)$, where $h_a$, $h_b$, and $h_c$ are the directed distances from $M$ to the corresponding sides of the reference triangle $ABC$.
• Figure 2: What are the homogeneous coordinates of midpoint $M_a$?
• Figure 3: Transforming the coordinates of points $P$ and $P'$ with positive (left) and negative (right) curvature. The coordinate origin point $T$ is the tangent point of the curved surface.
• Figure 4: A cross section through its center $O$ of an embedded sphere with radius $\sqrt{1/K}$. Point $M$ is a triangle center, and point $F$ is the foot of a perpendicular drawn on the sphere from the center to an edge.
• Figure 5: The point $M$ is the median point of triangle $ABC$.