Simplicial subdivision of simplices of arbitrary dimension in spaces of constant curvature with bounded quality

2026-07-07

AI Summary

This work addresses the problem of generating high-quality simplicial subdivisions in constant-curvature spaces of arbitrary dimension, ensuring that geometric quality measures—such as fatness—of all resulting subsimplices are uniformly bounded below by a positive constant. The authors extend prior two-dimensional results to arbitrary dimensions by introducing a novel strategy that combines Freudenthal’s Euclidean triangulation with radial projection, carefully adapted to the intrinsic geometry of constant-curvature spaces. This approach yields a unified refinement scheme applicable across spherical, Euclidean, and hyperbolic geometries, simultaneously preserving geometric fidelity and guaranteeing a strict positive lower bound on element quality.

Abstract

In 1942, Freudenthal showed that a simplex in Euclidean space can be subdivided such that the quality (well-shapedness of the simplex, quantified in terms of e.g. fatness) of the simplices in the subdivision is lower bounded. This answered a question of Brouwer. Recently, Brunck discussed the same problem for simplices in two-dimensional spaces of constant curvature and provided a closely related construction. In this paper we generalize Brunck's result to arbitrary dimensional spaces of constant curvature by combining Freudenthal's construction and radial projection. We contrast this approach with Brunck's construction.

Problem

• simplicial subdivision
• constant curvature
• simplex quality
• fatness
• arbitrary dimension

Innovation

• simplicial subdivision
• constant curvature spaces
• bounded quality
• Freudenthal's construction
• radial projection