Burning or collapsing the medial axis is unstable
Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing the medial axis is unstable. La Matematica.
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Author
Chambers, Erin Wolf;
Fillmore, Christopher DISTA;
Stephenson, Elizabeth RISTA
;
Wintraecken, MathijsISTA 


Corresponding author has ISTA affiliation
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Abstract
The medial axis of a set consists of the points in the ambient space without a unique closest point in the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a skeleton topologically equivalent to the original set. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities, various prunings of the medial axis have been proposed in the computational geometry community. Here, we examine one type of pruning, called burning. Because of the good experimental results it was hoped that the burning method of simplifying the medial axis would be stable. In this work, we show a simple example that dashes such hopes. Based on Bing’s house with two rooms, we demonstrate an isotopy of a shape where the medial axis goes from collapsible to non-collapsible. More precisely, we consider the standard deformation retract from the closed ball to Bing’s house with two rooms, but stop just short of the point where Bing’s house becomes two dimensional. This way we obtain an isotopy from the 3-ball to a thickened version of Bing’s house. Under this isotopy, the medial axis goes from collapsible to non-collapsible. We stress that this isotopy can be made generic, in the sense of singularity theory, as developed by Arnol’d and Thom.
Publishing Year
Date Published
2025-08-25
Journal Title
La Matematica
Publisher
Springer Nature
Acknowledgement
We thank André Lieutier, David Letscher, Ellen Gasparovic, Kathryn Leonard, and Tao Ju for early discussions on this work. We also thank Lu Liu, Yajie Yan, and Tao Ju for sharing code to generate the examples. We further thank Abigail Thompson for discussion on the conjecture and James Damon for sharing his insight in singularity theory. We thank the reviewers for their detailed reviews, which helped to improve the exposition.
Open access funding provided by Institute of Science and Technology (IST Austria). Partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’ and the European Research Council (ERC), grant no. 788183, ‘Alpha Shape Theory Extended’. The first author was supported in part by the National Science Foundation through grants DBI-1759807, CCF-1907612, and CCF-2444309. The fourth author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-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 Université Côte d’Azur, ANR-15-IDEX-01.
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Cite this
Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Burning or collapsing the medial axis is unstable. La Matematica. 2025. doi:10.1007/s44007-025-00170-0
Chambers, E. W., Fillmore, C. D., Stephenson, E. R., & Wintraecken, M. (2025). Burning or collapsing the medial axis is unstable. La Matematica. Springer Nature. https://doi.org/10.1007/s44007-025-00170-0
Chambers, Erin Wolf, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Burning or Collapsing the Medial Axis Is Unstable.” La Matematica. Springer Nature, 2025. https://doi.org/10.1007/s44007-025-00170-0.
E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Burning or collapsing the medial axis is unstable,” La Matematica. Springer Nature, 2025.
Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing the medial axis is unstable. La Matematica.
Chambers, Erin Wolf, et al. “Burning or Collapsing the Medial Axis Is Unstable.” La Matematica, Springer Nature, 2025, doi:10.1007/s44007-025-00170-0.
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