Symmedians as Hyperbolic Barycenters

TL;DR

This work introduces the hyperbolic barycenter $S_ extbf{P}$ of an ideal polygon as a natural hyperbolic analogue of the Euclidean triangle symmedian, with explicit coordinates defined by Hamiltonians $I_ extbf{P}, J_ extbf{P}, K_ extbf{P}$ and the key relation $S_ extbf{P}= igl(rac{I_ extbf{P}-K_ extbf{P}}{I_ extbf{P}+K_ extbf{P}}, rac{2J_ extbf{P}}{I_ extbf{P}+K_ extbf{P}}igr)$. The authors prove that $S_ extbf{P}$ minimizes the sum of hyperbolic sines of distances to the polygon’s sides, generalizing the classical symmedian’s optimality, and extend this notion to all ideal polygons via an Interpolation Lemma and Archimedean properties. They develop moduli spaces of ideal $n$-gons with fixed $S$, analyze explicit cases for triangles and quadrilaterals in terms of Poncelet conics, and explore the relation to harmonic polygons, showing that for harmonic polygons $S_ extbf{P}$ coincides with the Euclidean least-squares point $L_ extbf{P}$ after deKleinization. Together, these results establish a robust hyperbolic barycentric framework with constructive geometry, moduli theory, and connections to classical Euclidean optimization.

Abstract

The symmedian point of a triangle enjoys several geometric and optimality properties, which also serve to define it. We develop a new dynamical coordinatization of the symmedian, which naturally generalizes to other ideal hyperbolic polygons beyond triangles. We prove that in general this point still satisfies analogous geometric and optimality properties to those of the symmedian, making it into a hyperbolic barycenter. We initiate a study of moduli spaces of ideal polygons with fixed hyperbolic barycenter, and of some additional optimality properties of this point for harmonic (and sufficiently regular) ideal polygons.

Figures (7)

• Figure 1: Construction of the symmedian point $S$ of the triangle $\Delta(ABC)$.
• Figure 2: Another minimality property of the Symmedian point, from both the Klein-Beltrami (left) and the Poincaré (right) points of view.
• Figure 3: The hyperbolic barycenter of an ideal quadrilateral is the intersection point of its diagonals as well as the intersection of the segments connecting symmedians of opposite triangles in each triangulation.
• Figure 4: Constructions of the hyperbolic barycenter of an ideal pentagon. Left: via the Archimedian property. Right: via the Interpolation Lemma.
• Figure 5: Constructions of the hyperbolic barycenter of an ideal hexagon. Left: via the Archimedian property. Right: via the Interpolation Lemma.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 4.1
• Corollary 4.2
• Remark 4.3