Lightlike and ideal tetrahedra

TL;DR

This paper unifies the description of lightlike and ideal tetrahedra across various 3D geometries using a generalized cross-ratio, and computes their volumes as generalizations of classical hyperbolic volume formulas.

Contribution

It introduces a unified framework for describing lightlike and ideal tetrahedra in multiple geometries via a generalized cross-ratio and relates dual tetrahedra to Danciger's generalized ideal tetrahedra.

Findings

Tetrahedra are characterized by a generalized cross-ratio in a 2D real algebra.

Dual tetrahedra correspond to Danciger's generalized ideal tetrahedra.

Volumes are computed as functions of edge lengths or dihedral angles, generalizing classical formulas.

Abstract

We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra that generalizes the complex numbers. Equivalently, tetrahedra with lightlike faces are determined by a pair of edge lengths and their duals by a pair of dihedral angles. We prove that the dual tetrahedra are precisely the generalized ideal tetrahedra introduced by Danciger. Finally, we compute the volumes of both types of tetrahedra as functions of their edge lengths or dihedral angles, obtaining generalizations of the Milnor-Lobachevsky volume formula of ideal hyperbolic tetrahedra.