Volume formula for a $\mathbb{Z}_2$-symmetric spherical tetrahedron through its edge lengths

TL;DR

This paper derives volume formulas and trigonometric identities for a symmetric spherical tetrahedron with a specific $Z_2$ symmetry, expressed through its edge lengths, in 3D spherical space.

Contribution

It introduces new volume formulas and identities for a $Z_2$-symmetric spherical tetrahedron based on its edge lengths, expanding geometric understanding.

Findings

Derived explicit volume formulas for the symmetric tetrahedron.

Established new trigonometric identities related to the tetrahedron.

Enhanced geometric tools for spherical tetrahedron analysis.

Abstract

The present paper considers volume formulae, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation through angle $\pi$ in the middle points of a certain pair of its skew edges.