Trigonometry of partially truncated triangles and tetrahedra

Abstract

The first main results of this note establish forms of the hyperbolic laws of cosines and sines for certain classes of quadrilaterals and pentagons in the hyperbolic plane, having at least one ideal vertex and right angles at non-ideal vertices, in which the length of a horocyclic cross-section at an ideal vertex plays the role filled by the dihedral angle in the usual versions of these laws. The second set of main results concern transversal length, meaning the distance from a designated internal edge to its opposite, of partially truncated tetrahedra in three-dimensional hyperbolic space whose non-truncated vertices are ideal. Transversal lengths of such tetrahedra are proved to depend only on the entire collection of internal edge lengths (interpreted at ideal vertices in terms of horospherical cross-sections), and bounds on these lengths are established. The case of ideal tetrahedra (no truncated vertices) is also considered. All main results are established using the unifying perspective of the hyperboloid model and Lorentzian geometry. A thorough introduction to this perspective is provided, with references as appropriate.