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Synthetic geometry in hyperbolic simplices

### Andrew Clickard and Barry Minemyer

Vol. 15 (2022), No. 5, 885–906
##### 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 $\left(\tau ,g\right)$. 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 $\kappa$.

##### Keywords
hyperbolic simplex, synthetic geometry, orthogonal projection, Gram matrix
##### Mathematical Subject Classification
Primary: 51K10
Secondary: 51M09, 51M25, 53A70