Synthetic geometry in hyperbolic simplices

Clickard, Andrew; Minemyer, Barry

doi:10.2140/involve.2022.15.885

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Synthetic geometry in hyperbolic simplices

Andrew Clickard and Barry Minemyer

Vol. 15 (2022), No. 5, 885–906

DOI: 10.2140/involve.2022.15.885

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 $(τ, g)$ . 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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Keywords

hyperbolic simplex, synthetic geometry, orthogonal projection, Gram matrix

Mathematical Subject Classification

Primary: 51K10

Secondary: 51M09, 51M25, 53A70

Milestones

Received: 10 February 2022

Accepted: 6 March 2022

Published: 3 March 2023

Communicated by Gaven Martin

Authors

Andrew Clickard
Department of Mathematical and Digital Sciences Bloomsburg University Bloomsburg, PA United States

Barry Minemyer
Department of Mathematical and Digital Sciences Bloomsburg University Bloomsburg, PA United States

Vol. 15, No. 5, 2022