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Synthetic geometry in hyperbolic simplices

Andrew Clickard and Barry Minemyer

Vol. 15 (2022), No. 5, 885–906

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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hyperbolic simplex, synthetic geometry, orthogonal projection, Gram matrix
Mathematical Subject Classification
Primary: 51K10
Secondary: 51M09, 51M25, 53A70
Received: 10 February 2022
Accepted: 6 March 2022
Published: 3 March 2023

Communicated by Gaven Martin
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