Hyperbolic Tetrahedron: Volume Calculation with Application to the Proof of the Schläfli Formula

Abstract

We propose a new approach to the problem of calculations of volumes in the Lobachevsky space, and we apply this method to tetrahedra. Using some integral formulas, we present an explicit formula for the volume of a tetrahedron in the function of the coordinates of its vertices as well as in the function of its edge lengths. Finally, we give a direct analitic proof of the famous Schläfli formula for tetrahedra.

Keywords

• пространство Лобачевского
• тетраэдр
• объем
• интегральная формула
• формула Шлефли