Simplicial Subdivision of Simplices of Arbitrary Dimension in Spaces of Constant Curvature with Bounded Quality

Jul 16, 2026

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In 1942, Freudenthal proved that a simplex in Euclidean space can be subdivided so that the quality of all simplices in the refinement is bounded from below, answering a question posed by Brouwer. Building on this line of work, Brunck developed a closely related construction for two-dimensional constant-curvature spaces. This paper extends Brunck’s result to arbitrary dimensions by combining Freudenthal’s construction with radial projection, and it also contrasts the resulting approach with Brunck’s method.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages19

Reading Time48 min

Abstract

Introduction

Historical background and motivation

Main idea and outline

Preliminaries

Quality measure (fatness)

Freudenthal–Kuhn triangulations

Radial projection and subdivision scheme

Comparison with Brunck

arXiv :2607 .0680 1v 1 [ cs .CG] 7 Jul 2026 SIMPLICIAL SUBDIVISION OF SIMPLICES OF ARBITRARY DIMENSION IN SPACES OF CONSTANT CURVATURE WITH BOUNDED QUALITY ∗ Jean-Daniel Boissonnat,† Hana Dal Poz Kouˇrimsk´a,‡ Arijit Ghosh,§ and Mathijs Wintraecken¶ 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. ∗ The research leading to these results has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 339025 GUDHI (Algorithmic Foundations of Geometry Understanding in Higher Dimensions) . The first author is further supported by the French government, through the 3IA Cˆote d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002 . The last author is/has been supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) M-3073, ANR grant StratMesh, ANR-24-CE48-1899, and the welcome package from IDEX of the Universit´e Cˆote d’Azur, ANR- 15-IDEX-01 . †Datashape, Inria centre Universit´e Cˆote d’Azur, France, [jean-daniel.boissonnat@inria.fr](jean-daniel.boissonnat@inria.fr) ‡University of Potsdam, Germany, [hana.dal.poz.kourimska@uni-potsdam.de](hana.dal.poz.kourimska@uni-potsdam.de)[ ](hana.dal.poz.kourimska@uni-potsdam.de)§ACM Unit, Indian statistical institute, Kolkata, India, [arijitiitkgpster@gmail.com](arijitiitkgpster@gmail.com)[ ](arijitiitkgpster@gmail.com)¶ Datashape, Inria centre Universit´e Cˆote d’Azur, France, [mathijs.wintraecken@inria.fr](mathijs.wintraecken@inria.fr) 1 Introduction Subdividing simplices while preserving quality is a longstanding problem, which dates back at least to Brouwer. In 1942, Freudenthal addressed Brouwer’s question and showed in [10] that simplices in Euclidean space can be subdivided without decreasing the quality. Also within computational geometry the question has a long history, see e.g. [1] . Recently, simplices of good quality have regained attention because of their importance to numerical accuracy in numerical partial differential equations [11] . Simplex subdivision with bounded quality in two-dimensional spaces of constant curvature has recently been investigated by Brunck [4] . In this paper, we show that using Freudenthal’s construction together with the radial projection suffices to prove that simplices in spaces of constant curvature of any dimension can be subdivided with lower bounded quality. Outline In Section 2 we introduce the main tools for our theorem—the notion of quality, the Freudenthal–Kuhn triangulation, and the radial projection. We also explain Freudenthal’s subdivision which is a key ingredient for our subdivision scheme. In Section 3 we use these tools to establish a subdivision scheme for any simplex in a space of constant curvature that ensures a lower bound on the quality of the simplices within the subdivision. Finally, in Section 4 we compare our approach to the construction of Brunck [4], shedding light on how their construction can be extended from the considered two-dimensional simplices to simplices of arbitrary (finite) dimension. We conclude the article with several remarks and open questions (Section 5) . 2 Preliminaries The goal of this paper is to show th

FAQs

What foundational problem does the paper address?

The paper addresses how to subdivide a simplex while maintaining a lower bound on the subdivision quality, building on Freudenthal’s 1942 result that answered a question of Brouwer.

How is “quality” of simplices quantified in this work?

Quality is quantified using fatness, defined as the simplex volume divided by the length of its longest edge raised to the d-th power.

What is the main technical contribution for higher dimensions?

The paper generalizes Brunck’s two-dimensional constant-curvature construction to arbitrary dimensions by combining Freudenthal–Kuhn triangulation with a radial projection technique.

Jul 16, 2026

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