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arXiv · 2107.05706

Synthetic Geometry in Hyperbolic Simplices

Andrew Clickard·Barry Minemyer
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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 $\kappa$.

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Andrew Clickard, Barry Minemyer. 2021-07-12. Synthetic Geometry in Hyperbolic Simplices. https://doi.org/10.2140/involve.2022.15.885

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