Synthetic Geometry in Hyperbolic Simplices

TL;DR

This paper develops geometric formulas for hyperbolic simplices that depend solely on edge lengths, enabling precise distance and projection calculations in hyperbolic and Euclidean simplices with constant curvature.

Contribution

It introduces new formulas for distances and projections in hyperbolic simplices based only on edge lengths, extending to arbitrary constant curvature simplices.

Findings

Derived distance formulas for hyperbolic simplices

Established projection formulas in hyperbolic and Euclidean simplices

Extended formulas to simplices with arbitrary constant curvature

Abstract

Let $\tau$ be an $n$-simplex and let $g$ be a metric on $\tau$ with constant curvature $\kappa$. The lengths that $g$ assigns to the edges of $\tau$, along with the value of $\kappa$, uniquely determine all of the geometry of $(\tau, g)$. In this paper we focus on hyperbolic simplices ($\kappa = -1$) and develop geometric formulas which rely only on the edge lengths of $\tau$. Our main results are distance and projection formulas in hyperbolic simplices, as well as a projection formula in Euclidean simplices. We also provide analogous formulas in simplices with arbitrary constant curvature $\kappa$.