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arXiv · 2107.05706

Synthetic Geometry in Hyperbolic Simplices

Andrew Clickard·Barry Minemyer
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Abstract

Let $τ$ be an $n$-simplex and let $g$ be a metric on $τ$ with constant curvature $κ$. The lengths that $g$ assigns to the edges of $τ$, along with the value of $κ$, uniquely determine all of the geometry of $(τ, g)$. In this paper we focus on hyperbolic simplices ($κ= -1$) and develop geometric formulas which rely only on the edge lengths of $τ$. 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 $κ$.

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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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