# Burning or Collapsing the Medial Axis is Unstable

Erin Wolf Chambers, Christopher Fillmore, Elizabeth Stephenson, Mathijs Wintraecken

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Erin Wolf Chambers, Christopher Fillmore, Elizabeth Stephenson, Mathijs Wintraecken. Burning or Collapsing the Medial Axis is Unstable. La Matematica, 2025, 10.1007/s44007-025-00170-0. hal-05225734

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# Burning or Collapsing the Medial Axis is Unstable

Erin Wolf Chambers¹ ID · Christopher Fillmore² ID · Elizabeth Stephenson²,⁴ ID · Mathijs Wintraecken³ ID

Received: 10 April 2024 / Revised: 15 July 2025 / Accepted: 21 July 2025
© The Author(s) 2025

## 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.

**Keywords** Medial axis · Burning · Stability theory · Singularity theory

**Mathematics Subject Classification** 68U05 · 65D18 · 58K40 · 58K60

<footnote>Extended author information available on the last page of the article</footnote>
# 1 Introduction

The (interior) medial axis is a skeleton that captures many geometric and topological features. Formally, the medial axis of a closed set $S$ consists of all points in the ambient space (usually $\mathbb{R}^d$) that do not have a unique closed point on $S$. In this paper, the ambient space will be $\mathbb{R}^d$.

As we will discuss, significant effort has gone into the study of the stability of the medial axis and its various prunings. In this paper, we investigate the stability of two similar prunings of the medial axis: collapses and burnings (defined below) of the medial axis. We show that these prunings do not yield a geometrically or topologically stable core.

## The introductions of the medial axis

The term *medial axis* was introduced by Blum [1], who suggested its use in the context of shape recognition. However, the concept predates Blum’s work; to our best knowledge, the concept was first probed by Erdős [2, 3]. In [4], Federer introduced the reach of a (closed) set $S \subset \mathbb{R}^d$ as the minimum of the distance from $S$ to the medial axis, $\text{ax}(S)$, and studied some properties of the (complement of the) medial axis. More recently, its closure has been reintroduced as the central set [5].

*Spines and cut loci* We note that the medial axis is closely related to both the cut locus and spine. The cut locus is the generalization of the medial axis to Riemannian manifolds [6]. A spine is a subset (subpolyhedra) to which the (triangulated) manifold with boundary can collapse; these fulfill a similar role in 3-manifold topology [7, 8].

## Stability of the medial axis

Unfortunately, the medial axis is limited in its use by its (topological) instability under small perturbations [9]. Here small is understood to be small with respect to the Hausdorff distance, $d_H(S, S')$. See Fig. 1 for a standard example of such an instability.

If we restrict ourselves to a smaller class of spaces and perturbations, there are stability results available: Chazal and Soufflet [10] proved that the medial axis is stable with respect to the Hausdorff distance under ambient diffeomorphisms, under strong conditions. Namely, Chazal and Soufflet assumed that the set of positive reach, $S$, is a $\mathbb{C}^2$ manifold and the distortion is a $\mathbb{C}^2$ diffeomorphism of $\mathbb{R}^b$. Federer [4] proved that the reach is stable under $\mathbb{C}^{1,1}$ diffeomorphisms of the ambient space, that is, a diffeomorphism of the ambient space whose derivative and the derivative of the inverse

<figure><img src="image_4.png" /><figcaption>**Fig. 1** Small perturbations (with respect to the Hausdorff distance) can lead to large perturbations of the medial axis</figcaption></figure>
are Lipschitz. Using Federer’s result Chazal and Soufflet’s work [10] was recently improved by Dal Poz Kouřimská, Lieutier, and Wintraecken to the medial axis of any closed set with bounded complement under $C^{1,1}$ diffeomorphisms [11]. The medial axis does capture the homotopy type of $S$ [12, 13].

With some extra information and in the smooth setting the medial axis contains much more information than just the homotopy type: if we know the medial axis of a shape and the distance to a smooth manifold (of codimension one) for each point on the cut locus (the radius function), we can recover the manifold, as noted by Wolter [13]. Damon [14–17] investigated how to recover the differential geometry of the manifold from the geometry of the medial axis and the radius function.

*Generic stability and singularities* For generic points on the medial axis of a manifold embedded in low dimensions, the geometry is somewhat stable: generically a point on the medial axis in three dimensions has two closest points, in which case the geometry of a neighbourhood (or more precisely the second fundamental form) can be described in terms of the geometry near the two closest points, see Van Manen [18], see also [19–21]. This means that (again generically) the geometry of the medial axis is locally stable.

The topology of the singularities of the medial axis can also be classified in a generic setting in low dimensions. More precisely, any point on the medial axis of a generic manifold has a neighbourhood such that the medial axis in this neighbourhood is diffeomorphic to sets in the classification. The classification of singularities of the medial axis is the same as for the cut locus, which is, roughly speaking, the medial axis for a single point on a manifold, see Fig. 2. This classification was found in a variety of settings by Mather [22], Buchner [23] and Yomdin [24] based on work by Arnold [25], Thom [6] and Looijenga [26] and later refined by Damon and Gaspарovic [27]. In particular, Chapter 13 of [27] extensively discusses genericity, and we refer to this chapter for full discussion on the topic. However, we briefly note here that the set is open and dense, in some sense. We refer the reader to Van Manen [28]

<figure><img src="image_5.png" /><figcaption>**Fig. 2** The singularities of the medial axis of a generic surface embedded in $\mathbb{R}^3$ (including the ‘non-singularity’, that is the surface). The double lines indicate open ends. The $A_i^j$ notation for the types of singularities was introduced by Arnol’d</figcaption></figure>
for a more in depth overview of this rich topic and its history, as well as to to work by Damon [29]. A finite classification of singularities is only possible in sufficiently low dimensions.<fnref n="1" />.

The generic transitions between singularities of the medial axis, called perestroikas, in a one parameter family isotopy between (generic) surfaces in $\mathbb{R}^3$ have been studied by Giblin and Kimia [33], based on earlier work by Bogaevsky [32, 34–36]. We represent the transitions given by Giblin and Kimia, and Bogaevsky in Fig. 3. In this work the $A_1^4$ transition plays a significant role.

### Prunings

Significant effort has gone into the simplification (pruning) of the medial axis. This was motivated by applications in graphics (where the medial axis is used as a skeleton, see the surveys [37, 38]), data reduction, shape recognition, and machine learning (see for example [39–46]). Various prunings of the medial axis have been proposed in many different settings [43, 45, 47–53]. See Fig. 4 for an illustration of some commonly used prunings and their pitfalls: the object angle, which is of historic importance in the community but can disconnect the medial axis, the $\lambda$-medial axis, which is used to compute a close approximation of the medial axis but which can truncate “thin” regions undesirably, and the burning method, which we consider in this work.

Lieutier and Wintraecken [54] recently showed that there is a pruning (the $(\lambda, \alpha)$-pruning) of the medial axis that is Hausdorff and Gromov-Hausdorff stable under

<figure><img src="image_6.png" /><figcaption>**Fig. 3** The generic transitions of singularities (perestroikas) of the medial axis in $\mathbb{R}^3$. Adapted from figures by Bogaevsky [32]. We use the same notation as [33], which in turn follows Arnol'd. As we'll see in Fig. 8, see also Fig. 13, it is the transition of type $A_1^4$ that causes the (interior) medial axis to become non-collapsible (during the deformation of the cube into a thickened Bing's house)</figcaption></figure>

<footnote n="1">¹In sufficiently high dimensions the classification of the singularities (or rather jets that in turn yield the singularities) does in general not yield finite distinct cases, but instead multiparameter families. For a small (one or two) number of parameters these families have been determined explicitly, see Arnold [65]. The situation is also more complicated in high dimensions, because of lack of stability in the smooth sense. More explicitly, Buchner [23, 30], see also [31, Theorem 4 (i) and (ii), page 737], that in dimension 7 or higher, there is no stability of the singularities of the cut locus in the $C^\infty$ sense only of in the $C^0$ sense. Topological stability follows from the work of Looijenga [26]</footnote>
<figure><img src="image_7.png" /><figcaption>**Fig. 4** Various pruning methods, from left to right: Object angles [48, 52], (circum)radius of the set of closest points [50] (the λ-medial axis, also used in our computation), and a burning method proposed in [46], with various undesirable features indicated. The value of the object angle, radius of the set of closest points, and burning time (referred to as erosion thickness in [46]) is indicated in colour on top. Reproduced from [46]</figcaption></figure>

Hausdorff perturbations of the input set. We recall that the Gromov-Hausdorff metric measures how different metric spaces are.

We note that there is a previous extended abstract overlapping some of this paper’s content, but lacking most of the proofs and details, which was published in the Multimedia Exposition workshop at the Symposium for Computational Geometry in 2022 [55]. This abstract also comprised a video animation of a deformation retract to better illustrate the results.

## 2 Burning Bing's House

The simplification which we focus on for this work is the burning of the medial axis [46, 56], which generalizes Blum’s original “grassfire” analogy for the medial axis.

*Burning and Bing's house* The burning of the medial axis removes the extremities of the medial axis by ‘starting a fire’ at the boundary of the medial axis which stops if the fire hits an obstacle, as illustrated in Fig. 5. Note that this illustration focuses on 2 dimensions, where the definition is simpler [56]. The generalization of burning to 3 dimensions is significantly more complex, as the singularites are no longer points. We refer the reader to [46] for the full definition and examples of this generalization, as the full details are not needed for our main result.
<figure><img src="image_8.png" /><figcaption>**Fig. 5** The fire front progression on a the medial axis (grey) of a curve (black). As the fire front (indicated by the red dot) hits an unburned junction, it stops. If the junction is already burned (with the colour indicating the burn time) the fire continues. As in Fig. 4 the burning time is indicated in colour</figcaption></figure>

Because of the good experimental results, it was conjectured that the burning method of simplification of the medial axis would be stable [43], i.e. no discontinuous jumps. In this work we show that this is not the case. The counter example is based on the standard deformation retraction from the closed ball to Bing's house with two rooms [57], which is a contractible but not collapsible two dimensional simplicial complex, see Fig. 6 and 7. Bing's house is not collapsible, as there is no boundary from which a collapsing sequence could begin.

*Deforming a cube into Bing's house* Bing constructed his house as a deformation retract from a solid cube, see Fig. 7. The deformation retraction goes as follows. During the entire construction, with the exception of the final step, the set will remain three dimensional. As mentioned we start with a solid cube. We then start by digging two corridors. Once we have reached the part of the solid cube that will because a room, we start hollowing out the room by protruding a rectangular cuboid into the room. This can be continued until we come near the corridor that goes to the opposite room. Of course we cannot interfere with the corridor, instead we engulf the corridor from both sides. Once we are past the corridor, we extend the hollowed space until it almost touches itself near the flap. We finally flatten the three dimensional set to two dimensions, creating Bing's house.

As indicated in Fig. 7 the deformation retraction of the solid cube to Bing's house can be done in a Piecewise Linear way. At every point before the final state where the set becomes two dimensional, the boundary is a Piecewise Linear 2-sphere. Because the boundary of the Piecewise Linear complex is a 2-sphere and all angles between adjacent faces of the Piecewise Linear complex are $\pi/2$, the boundary complex can be made $C^{1,1}$ by replacing the edges with parts of cylinders and the vertices with parts of spheres. Such $C^{1,1}$ manifolds can be smoothed, even without decreasing the reach, see [58]. This means in particular that a smooth version of the deformation retraction from the smoothed solid cube to a thickening of Bing's house exists.

*Deforming and computation* Because pieces of planes, cylinders, and sphere have relatively simple algebraic descriptions, it is in principle possible to calculate the medial axis using algebraic techniques, see e.g. [59]. However, because of our somewhat limited computational resources, we use the $\lambda$-medial axis to approximate and

<figure><img src="image_9.png" /></figure>
visualize this reformation. As mentioned, a variant of the $\lambda$-medial axis is stable in the Hausdorff and Gromov-Hausdorff sense if we take a sample that is close to the underlying set in terms of the Hausdorff distance (under technical assumptions) [54], but that does not mean that it captures the smooth structure. However, as we will see in Figs. 12 and 13, the parts of the medial axis that participate in the Perestroikas and hence are of the most interest all have large $\lambda$, that is, the smallest enclosing sphere of the closest point on the input set to such an interesting point on the medial axis is large. Hence, the $\lambda$-medial axis illustration captures the relevant behaviour of the exact medial axis.

*Deforming Bing's house into a collapsible complex* Before we go into the main statement, we consider a deformation of Bing's house which makes it collapsible. This deformation will be mirrored in the medial axis in our construction and is depicted in Fig. 8; see Fig. 6 for the nomenclature. In this construction we cut a flap open so that the room no longer completely runs around the corridor. This cutting exposes an edge of one of the walls of the corridor and path that goes from the edge to the bottom room. We can use this edge to collapse along the path into the bottom room, then the bottom room itself, and from this the rest of Bing's house.

**Main result**

We are now going to use the construction of Bing's house from a solid cube, but stop just before the set becomes two dimensional. This gives an isotopy of the boundary of the cube, namely $S^2$, such that the (interior$^2$) medial axis transitions from being collapsible and burnable to a point into Bing's house itself.

The precise result is the following:

<figure><img src="image_10.png" /><figcaption>**Fig. 6** The various parts of Bing's house indicated</figcaption></figure>

<footnote n="2">²The sphere $S^2$ divides the $\mathbb{R}^3$ into a compact or interior part and a non-compact or exterior part. By the interior medial axis we mean the part of the medial axis that lies on the inside of the compact or interior part.</footnote>
**Theorem 2.1** There exists a smooth ambient isotopy $H_t : [0, 1] \times \mathbb{S}^2 \to \mathbb{R}^3$ such that:

* The (interior) medial axis $\operatorname{ax}(H_0(\mathbb{S}^2))$ is collapsible/burnable to a single point.

* The (interior) medial axis $\operatorname{ax}(H_1(\mathbb{S}^2))$ is Bing's house and is therefore non-collapsible/cannot burn (is fireproof).

* The burning of $\operatorname{ax}(H_t(\mathbb{S}^2))$ is not continuous in $t$ with respect to the Hausdorff distance and Gromov-Hausdorff (as well as the Fréchet distance which is infinite).

* The topology of the burned (interior) medial axis changes from a point to Bing's house with two rooms at a single $t_0 \in [0, 1]$.

* The isotopy $H_t$ can be chosen to be generic in the sense of singularity theory as developed by Arnold<fnref n="3" /> and Thom [25], see in particular [33].

*Proof* Bing constructed his house as a deformation retract from a solid cube, see Fig. 7. The isotopy of the sphere we consider is the boundary of this deformation. However instead of reducing to a two dimensional object we forego the final step so that at any point in the deformation the set remains a topological (solid) ball and its boundary a sphere. The end point of this deformation is a thickened version of Bing's house.

<figure><img src="image_11.png" /><figcaption>**Fig. 7** The deformation retraction of a solid cube (topological ball) to Bing's house. In the final frame we show the smoothed version of a thickened Bing's house used in the computation</figcaption></figure>

‘Arnold’, as opposed to ‘Arnol’d’; the latter is closer to the Russian pronunciation, but Vladimir Arnold preferred the former (it is used in numerous translations of his books into English), and we use it throughout.” Here the quotation marks indicate a direct quotation from [60], which perfectly expresses our position.

<footnote n="3">³Most translations into English would perhaps cite this author as Arnol'd. Following [60], "we use</footnote>
We will only consider the medial axis in the interior of the sphere and not the exterior. The medial axis of a thickened version of Bing's house is Bing's house itself. The deformation is depicted in Fig. 12. The essential topological change only happens near the end of the deformation when the rooms wrap around their respective corridors, see Fig. 13. At this moment the bisector between the corridor and the wall disappears and is replaced by the bisector between the two parts of the room that are wrapping around the corridor and the medial axis becomes non-collapsible.

The transition as depicted in Fig. 12 is not generic, it is in fact quite symmetrical and at the critical transition, see Fig. 13, there is an entire edge in the medial axis for which there are 4 nearest points on the boundary of the thickened Bing's house. For each point on the edge, the closest points are a point on the exterior wall, a point on the corridor and two points on either side that lie on the expanding top room.

We can perturb or change the configuration slightly as follows in order to make it generic: Just before the time where the edge(s) in the medial axis for which there are 4 nearest points on the boundary of the thickened Bing's house is created, we can deform a small part of the thickened Bing's house. These edges come in pairs, one for the top room one for the bottom. For each of the edges the four closest points on the thickening of Bing's house lie on two sides of the growing room: one near the corridor, and one on the outside wall. This gives rise to 4 curves on the boundary of the thickened Bing's house. These curves are in fact straight lines as the thickening of Bing's house is locally flat in this neighbourhood. In the middle of these curves on the wall and near the corridor, we deform the thickening of Bing's house so that it becomes locally spherical (with a large radius) in a small neighbourhood of the edge. For the curves on the growing rooms we deform in the same manner, that is, we make it locally spherical, but with a much smaller radius. The construction has been illustrated in Fig. 9.

For all points $x$ near the centre of the configuration depicted in Fig. 9, the points $p \in \pi(x)$ in the closest point projection $\pi$ of $x$, lie on a part of a sphere. Because spheres and the distance function to spheres are simple to describe, we can compute the medial axis using isosurfacing techniques. We can perform this calculation for any time $t$ near the time of the transition. Here the two walls of the expanding top (or bottom) room become closer, while the distance between the corridor and exterior wall remains roughly the same. By focussing on the spherical parts, which locally control the transition, we see this is equivalent to two small spheres moving closer and two spheres remaining at the same distance, see Fig. 10. Figures 10 and 11, depict the results of these calculations, which indeed show the appearance of the perestroika or a transition between singularities of type $A_1^4$. Figure 8 offers a global view of what happens to the topology.

Figure 8 also helps to explain why the medial axis becomes collapsible. By the perestroika, the entire length of the corridor is exposed to the outside. We can therefore start collapsing/burning from the top of the corridor. We then continue to collapse/burn along parts of the corridor and separator that have been exposed by the splitting of the flap. We then can continue into the walls and floor of the bottom room, removing those. This exposes the bottom of the corridor leading to the top room, and we can collapse/burn this too. Finally we can collapse/burn the top room itself to a point.
<figure><img src="image_12.png" /><figcaption>**Fig. 8** This deformation (top six panels), which cuts a flap open, makes Bing's house collapsible. The topology of this transition (in the neighbourhood of the point where the flap disappears) is precisely that of the type $A_1^4$. The transition of type $A_1^4$ is depicted in the bottom three panels; we have coloured the strata to indicate the correspondence with the construction directly above</figcaption></figure>

<figure><img src="image_13.png" /></figure>

Because Bing's house cannot be pruned or collapsed we immediately have the following corollary:

**Corollary 2.2** *Collapsing or pruning the medial axis of a domain such that it becomes one-dimensional, as proposed in e.g. [49], is not always possible, even if the boundary of the domain is a smooth sphere.*

**2.1 Video**

The frames of the deformation illustrated in Figs. 7, 12, and 13 are taken from the multimedia contribution presented at the Symposium on Computational Geometry [55], the video can be found at https://youtu.be/CFmFP6CHVEk?si=rN3ZNBUPSG

<figure><img src="image_14.png" /></figure>
<figure><img src="image_15.png" /><figcaption>Fig. 9 The deformation near the critical transition (exaggerated)</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 10 The medial axis (blue) of 4 spheres (pink) two of with a large radius and two with a small radius</figcaption></figure>

aYe06U. These animations were made using the $\lambda$-medial axis (see https://github.com/cdfillmore/lambda_medial_axis) and the open source-software Blender [62]. Here $\lambda$ is chosen very small to ensure that the $\lambda$-medial axis is a good approximation of the medial axis.

## 3 Conclusion and Open Question

In this paper we constructed an isotopy of the sphere such that the interior medial axis transitions from collapsible to non-collapsible. The exterior medial axis throughout this isotopy remains collapsible (by which we mean after compactifying the ambient space to $S^3$). This automatically leads some to interesting questions:
<figure><img src="image_17.png" /><figcaption>**Fig. 11** A zoom in on the centre of the medial axis of (blue) of 4 spheres (pink) two of with a large radius (9 + 1/20) and two with a small radius (1/20), with centres (0, ±(9 + 1/10), 0) and (±(1/10 + t), 0, 0), for $400t = -4, -3, -2, -1, 0, 1$. Computed and visualized using Wolfram Mathematica 12.0. Because the bisectors between various spheres intersect transversally, we could have alternatively used the algorithm of [61], which comes with guarantees, but no visualisation tools that any of the authors are able to use</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 12** The evolution of the interior medial axis (yellow) as the smoothed solid cube is deformed into a thickened version of Bing's house (blue/purple). The smoothed solid cubes in these five frames coincide with the unsmoothed solid cubes in the first five frames of Fig. 7. The exact moment at which the medial axis becomes non-collapsible lies between frames four and five and is depicted explicitly in Fig. 13</figcaption></figure>

<figure><img src="image_19.png" /></figure>
**Fig. 13** The critical transition of the medial axis. There are points on the medial axis equidistant to the two parts of the room that wrap around the corridor, the corridor itself, and the exterior wall, which can be avoided by a small perturbation. This transition occurs between frames 4 and 5 of Fig. 7

<figure><img src="image_20.png" /></figure>

**Question 1** Does there exist an embedding of $S^2$ in $S^3$ where it is not possible to burn/partially collapse either connected component of the medial axis of $S^2$ at all?

Here the medial axis should be understood as the cut locus medial axis. The cut locus medial axis of a closed subset $S$ of a manifold $M$ is the set of points $p$ in the manifold $M$ for which there is no unique (minimizing) geodesic from $p$ to $S$ with length equal to $d_M(p, S)$, where $d_M$ denotes the distance on $M$. We refer to [63, Appendix F] for details. Also in this setting the medial axis captures the homotopy type [64], so that there are two connected components, explaining the ‘either’ in the question. We’ll refer to one of the connected components as the exterior and one as the interior, as before, even though this choice is arbitrary (unless $S^3$ is explicitly described as the one-point compactification of Euclidean space).

We suspect that this question can be answered in the negative. We can also imagine that we can use the burning of the exterior medial axis to guide an isotopy of the sphere. See Fig. 14 for an example of an isotopy guided by burning. For the thickened version of Bing’s house, an isotopy guided by the burning of the medial axis would yield the inverse of the standard construction of Bing’s house (which we described above and is depicted in Fig. 7), see Fig. 15 for a depiction of how the burning would guide the isotopy.

This leads us to the following stronger conjecture:

**Conjecture 3.1** *Given any embedding of $S^2$ in $S^3$, there exists a series of isotopies of $S^2$ based on successive burnings of both the interior and exterior medial axes, which yields the standard symmetrical embedding of the 2-sphere in $S^3$.*

We point out that some work on the correspondence between the singularities of the interior and exterior medial axes has already been done [27].
<figure><img src="image_21.png" /><figcaption>**Fig. 14** The medial cactus: a two dimensional example of an isotopy guided by the burning of the medial axis. In this case the isotopy goes from a cactus shape into the (round) circle</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. 15** In this figure we indicate an isotopy guided by the burning of the medial axis in the top room of a thickened version of Bing's house. We see how the room deforms from enveloping the corridor (to the bottom room) to a circular remnant. After this deformation, the shape is a smoothed version of the second frame of Fig. 7. The corridor in this frame can again be isotoped away, using the burning of the medial axis as a guide</figcaption></figure>

**Acknowledgements** 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.

**Funding** 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.

**Declarations**

**Conflict of interest** The authors declare they have no Conflict of interest.

<figure><img src="image_23.png" /></figure>
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## Authors and Affiliations

Erin Wolf Chambers<fnref n="1" /> ID · Christopher Fillmore<fnref n="2" /> ID · Elizabeth Stephenson<fnref n="2" />,<fnref n="4" /> ID · Mathijs Wintraecken<fnref n="3" /> ID

<figure><img src="image_24.png" /></figure>

Christopher Fillmore
christopher.fillmore@ist.ac.at

Erin Wolf Chambers
echambe2@nd.edu

Elizabeth Stephenson
elizasteprene@gmail.com

Mathijs Wintraecken
mathijs.wintraecken@inria.fr

<figure><img src="image_25.png" /></figure>

<footnote n="1">¹ Department of Computer Science and Engineering, University of Notre Dame, 384 Fitzpatrick Hall of Engineering, Notre Dame, IN 46556, USA</footnote>

<footnote n="2">² Edelsbrunner group, IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria</footnote>

<footnote n="3">³ Centre Inria d'Université Côte d'Azur, Université Côte d'Azur, Inria, 2004 Rte des Lucioles, 06902 Valbonne, France</footnote>

<footnote n="4">⁴ Orteliu, Oslo, Norway</footnote>