---
OA_place: publisher
_id: '21021'
abstract:
- lang: eng
text: This thesis examines how geometry and topology intersect in the representation,
transformation, and analysis of complex shapes. It considers how continuous manifolds
relate to their discrete analogues, how topological structures evolve in persistence
vineyards, and how tools from topological data analysis can illuminate problems
in mathematical physics. Central to this exploration is the question of how structure,
both geometric and topological, persists or changes under approximation, sampling,
or deformation. The work develops new approaches to skeletal and grid-based representations
of surfaces, reveals the full expressive capacity of persistence vineyards, and
applies topological methods to the longstanding problem of equilibria in electrostatic
fields. These threads braid together into a broader understanding of how topology
and geometry inform one another across theory, computation, and application.
acknowledged_ssus:
- _id: M-Shop
- _id: ScienComp
acknowledgement: "The research presented in this thesis was funded by the DFG Collaborative
Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’.\r\n"
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
citation:
ama: Fillmore CD. Braiding geometry and topology to study shapes and data. 2026.
doi:10.15479/AT-ISTA-21021
apa: Fillmore, C. D. (2026). Braiding geometry and topology to study shapes and
data. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-21021
chicago: Fillmore, Christopher D. “Braiding Geometry and Topology to Study Shapes
and Data.” Institute of Science and Technology Austria, 2026. https://doi.org/10.15479/AT-ISTA-21021.
ieee: C. D. Fillmore, “Braiding geometry and topology to study shapes and data,”
Institute of Science and Technology Austria, 2026.
ista: Fillmore CD. 2026. Braiding geometry and topology to study shapes and data.
Institute of Science and Technology Austria.
mla: Fillmore, Christopher D. Braiding Geometry and Topology to Study Shapes
and Data. Institute of Science and Technology Austria, 2026, doi:10.15479/AT-ISTA-21021.
short: C.D. Fillmore, Braiding Geometry and Topology to Study Shapes and Data, Institute
of Science and Technology Austria, 2026.
corr_author: '1'
date_created: 2026-01-20T21:38:40Z
date_published: 2026-01-21T00:00:00Z
date_updated: 2026-04-07T11:42:49Z
day: '21'
ddc:
- '514'
- '516'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
- _id: UlWa
doi: 10.15479/AT-ISTA-21021
file:
- access_level: open_access
checksum: 4c0889130095c31d4e5088c5b8dfd607
content_type: application/pdf
creator: cfillmor
date_created: 2026-01-26T19:44:46Z
date_updated: 2026-01-30T11:40:09Z
file_id: '21046'
file_name: 2025_Fillmore_Christopher_Thesis.pdf
file_size: 55954297
relation: main_file
- access_level: closed
checksum: d69afb71d82ab98f856886126ee7303a
content_type: application/x-zip-compressed
creator: cfillmor
date_created: 2026-01-26T19:46:20Z
date_updated: 2026-01-26T19:46:20Z
file_id: '21047'
file_name: Thesis.zip
file_size: 166080788
relation: source_file
file_date_updated: 2026-01-30T11:40:09Z
has_accepted_license: '1'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: '122'
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
record:
- id: '20260'
relation: part_of_dissertation
status: public
- id: '21050'
relation: part_of_dissertation
status: public
- id: '21051'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
title: Braiding geometry and topology to study shapes and data
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2026'
...
---
OA_place: repository
_id: '21051'
abstract:
- lang: eng
text: 'In this work, we introduce and study what we believe is an intriguing and,
to the best of our knowledge, previously unknown connection between two areas
in computational topology, topological data analysis (TDA) and knot theory. Given
a function from a topological space to $\mathbb{R}$, TDA provides tools to simplify
and study the importance of topological features: in particular, the $l^{th}$-dimensional
persistence diagram encodes the $l$-homology in the sublevel set as the function
value increases as a set of points in the plane. Given a continuous one-parameter
family of such functions, we can combine the persistence diagrams into an object
known as a vineyard, which track the evolution of points in the persistence diagram.
If we further restrict that family of functions to be periodic, we identify the
two ends of the vineyard, yielding a closed vineyard. This allows the study of
monodromy, which in this context means that following the family of functions
for a period permutes the set of points in a non-trivial way. In this work, given
a link and value $l$, we construct a topological space and periodic family of
functions such that the closed $l$-vineyard contains this link. This shows that
vineyards are topologically as rich as one could possibly hope. Importantly, it
has at least two immediate consequences: First, monodromy of any periodicity can
occur in a $l$-vineyard, answering a variant of a question by [Arya et al 2024].
To exhibit this, we also reformulate monodromy in a more geometric way, which
may be of interest in itself. Second, distinguishing vineyards is likely to be
difficult given the known difficulty of knot and link recognition, which have
strong connections to many NP-hard problems.'
article_processing_charge: No
arxiv: 1
author:
- first_name: Erin
full_name: ' Chambers, Erin'
last_name: ' Chambers'
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. Braiding vineyards.
arXiv. doi:10.48550/ARXIV.2504.11203
apa: Chambers, E., Fillmore, C. D., Stephenson, E. R., & Wintraecken, M. (n.d.).
Braiding vineyards. arXiv. https://doi.org/10.48550/ARXIV.2504.11203
chicago: Chambers, Erin, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs
Wintraecken. “Braiding Vineyards.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2504.11203.
ieee: E. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Braiding
vineyards,” arXiv. .
ista: Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. Braiding vineyards.
arXiv, 10.48550/ARXIV.2504.11203.
mla: Chambers, Erin, et al. “Braiding Vineyards.” ArXiv, doi:10.48550/ARXIV.2504.11203.
short: E. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, ArXiv (n.d.).
corr_author: '1'
date_created: 2026-01-27T14:41:44Z
date_published: 2026-01-02T00:00:00Z
date_updated: 2026-04-07T11:42:48Z
day: '02'
department:
- _id: HeEd
doi: 10.48550/ARXIV.2504.11203
external_id:
arxiv:
- '2504.11203'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2504.11203
month: '01'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
record:
- id: '21056'
relation: later_version
status: public
- id: '21021'
relation: dissertation_contains
status: public
status: public
title: Braiding vineyards
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '21056'
abstract:
- lang: eng
text: "In this work, we introduce and study what we believe is an intriguing, and,
to the best of our knowledge, previously unknown connection between two fundamental
areas in computational topology, namely topological data analysis (TDA) and knot
theory. Given a function from a topological space to ℝ, TDA provides tools to
simplify and study the importance of topological features: in particular, the
\U0001D459^\U0001D461ℎ-dimensional persistence diagram encodes the topological
changes (or \U0001D459-homology) in the sublevel set as the function value increases
into a set of points in the plane. Given a continuous one parameter family of
such functions, we can combine the persistence diagrams into an object known as
a vineyard, which tracks the evolution of points in the persistence diagram as
the function changes. If we further restrict that family of functions to be periodic,
we identify the two ends of the vineyard, yielding a closed vineyard. This allows
the study of monodromy, which in this context means that following the family
of functions for a period permutes the set of points in a non-trivial way. Recent
work has studied monodromy in the directional persistent homology transform, demonstrating
some interesting connections between an input shape and monodromy in the persistent
homology transform for 0-dimensional homology embedded in ℝ^2.\r\nIn this work,
given a link and a value \U0001D459, we construct a topological space (based on
the given link) and periodic family of functions on this space (based on the Euclidean
distance function), such that the closed \U0001D459-vineyard contains this link.
This shows that vineyards are topologically as rich as one could possibly hope,
suggesting many future directions of work. Importantly, it has at least two immediate
consequences we explicitly point out:\r\n1.\tMonodromy of any periodicity can
occur in a \U0001D459-vineyard for any \U0001D459. This answers a variant of a
question by Arya and collaborators. To exhibit this as a consequence of our first
main result we also reformulate monodromy in a more geometric way, which may be
of interest in itself.\r\n2.\tTopologically distinguishing closed vineyards is
likely to be difficult (from a complexity theory as well as from a practical perspective)
because of the difficulty of knot and link recognition, which have strong connections
to many NP-hard problems."
acknowledgement: We thank the reviewers for both SODA and ATMCS for their comments,
whichimproved the exposition. We thank Kate Turner for discussion and Clément Maria
for pointing out thatAlexander’s theorem was already (well) known. Mathijs Wintraecken
would like to express his gratitude tothe administrative support he received from
University of Notre Dame during his visit and from Sophie Honnoratand Stephanie
Verdonck at Inria in general.This work has been supported by the ANR grant StratMesh,
ANR-24-CE48-1899, by NSF award 2444309, andthe welcome package from IDEX of the
Université Côte d’Azur, ANR-15-IDEX-01.
article_processing_charge: No
arxiv: 1
author:
- first_name: Erin W.
full_name: Chambers, Erin W.
last_name: Chambers
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: 'Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Braiding Vineyards.
In: Green Larsen K, Saha B, eds. Proceedings of the 2026 Annual ACM-SIAM Symposium
on Discrete Algorithms. Philadelphia, PA, United States: Society for Industrial
and Applied Mathematics; 2026:6240-6263. doi:10.1137/1.9781611978971.225'
apa: 'Chambers, E. W., Fillmore, C. D., Stephenson, E. R., & Wintraecken, M.
(2026). Braiding Vineyards. In K. Green Larsen & B. Saha (Eds.), Proceedings
of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 6240–6263).
Philadelphia, PA, United States: Society for Industrial and Applied Mathematics.
https://doi.org/10.1137/1.9781611978971.225'
chicago: 'Chambers, Erin W., Christopher D Fillmore, Elizabeth R Stephenson, and
Mathijs Wintraecken. “Braiding Vineyards.” In Proceedings of the 2026 Annual
ACM-SIAM Symposium on Discrete Algorithms, edited by Kasper Green Larsen and
Barna Saha, 6240–63. Philadelphia, PA, United States: Society for Industrial and
Applied Mathematics, 2026. https://doi.org/10.1137/1.9781611978971.225.'
ieee: 'E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Braiding
Vineyards,” in Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete
Algorithms, K. Green Larsen and B. Saha, Eds. Philadelphia, PA, United States:
Society for Industrial and Applied Mathematics, 2026, pp. 6240–6263.'
ista: 'Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2026.Braiding Vineyards.
In: Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms.
, 6240–6263.'
mla: Chambers, Erin W., et al. “Braiding Vineyards.” Proceedings of the 2026
Annual ACM-SIAM Symposium on Discrete Algorithms, edited by Kasper Green Larsen
and Barna Saha, Society for Industrial and Applied Mathematics, 2026, pp. 6240–63,
doi:10.1137/1.9781611978971.225.
short: E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, in:, K. Green
Larsen, B. Saha (Eds.), Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete
Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA,
United States, 2026, pp. 6240–6263.
date_created: 2026-01-28T12:58:16Z
date_published: 2026-01-07T00:00:00Z
date_updated: 2026-02-16T08:06:23Z
day: '07'
department:
- _id: HeEd
doi: 10.1137/1.9781611978971.225
editor:
- first_name: Kasper
full_name: Green Larsen, Kasper
last_name: Green Larsen
- first_name: Barna
full_name: Saha, Barna
last_name: Saha
external_id:
arxiv:
- '2504.11203'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2504.11203
month: '01'
oa: 1
oa_version: Preprint
page: 6240-6263
place: Philadelphia, PA, United States
publication: Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms
publication_identifier:
eisbn:
- '9781611978971'
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
related_material:
record:
- id: '21051'
relation: earlier_version
status: public
status: public
title: Braiding Vineyards
type: book_chapter
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2026'
...
---
OA_place: repository
OA_type: green
_id: '21931'
abstract:
- lang: eng
text: In 1873, James C. Maxwell conjectured that the electric field generated by
n point charges in generic position has at most (n-1)^2 isolated zeroes. The first
(nonoptimal) upper bound was only obtained in 2007 by Gabrielov, Novikov, and
Shapiro, who also posed two additional interesting conjectures. In this article,
we give the best upper bound known to date on the number of zeroes of the electric
field, and construct a counterexample to Conjecture 1.8 by Gabrielov, Novikov,
and Shapiro that the number of equilibria cannot exceed those of the distance
function defined by the unit point charges. Finally, we note that it is quite
possible that Maxwell's quadratic upper bound is not tight, so it is prudent to
find lower bounds. Hence, we also explore examples and construct configurations
of charges achieving the highest ratios of the number of electric field zeroes
by point charges found to this day.
article_number: e70163
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Goncalo
full_name: Oliveira, Goncalo
id: 58abbde8-f455-11eb-a497-98c8fd71b905
last_name: Oliveira
citation:
ama: Edelsbrunner H, Fillmore CD, Oliveira G. Counting equilibria of the electrostatic
potential. Proceedings of the London Mathematical Society. 2026;132(5).
doi:10.1112/plms.70163
apa: Edelsbrunner, H., Fillmore, C. D., & Oliveira, G. (2026). Counting equilibria
of the electrostatic potential. Proceedings of the London Mathematical Society.
Wiley. https://doi.org/10.1112/plms.70163
chicago: Edelsbrunner, Herbert, Christopher D Fillmore, and Goncalo Oliveira. “Counting
Equilibria of the Electrostatic Potential.” Proceedings of the London Mathematical
Society. Wiley, 2026. https://doi.org/10.1112/plms.70163.
ieee: H. Edelsbrunner, C. D. Fillmore, and G. Oliveira, “Counting equilibria of
the electrostatic potential,” Proceedings of the London Mathematical Society,
vol. 132, no. 5. Wiley, 2026.
ista: Edelsbrunner H, Fillmore CD, Oliveira G. 2026. Counting equilibria of the
electrostatic potential. Proceedings of the London Mathematical Society. 132(5),
e70163.
mla: Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.”
Proceedings of the London Mathematical Society, vol. 132, no. 5, e70163,
Wiley, 2026, doi:10.1112/plms.70163.
short: H. Edelsbrunner, C.D. Fillmore, G. Oliveira, Proceedings of the London Mathematical
Society 132 (2026).
corr_author: '1'
date_created: 2026-05-31T22:02:13Z
date_published: 2026-05-01T00:00:00Z
date_updated: 2026-06-02T09:24:18Z
day: '01'
department:
- _id: HeEd
- _id: TaHa
doi: 10.1112/plms.70163
external_id:
arxiv:
- '2501.05315'
intvolume: ' 132'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2501.05315
month: '05'
oa: 1
oa_version: Preprint
publication: Proceedings of the London Mathematical Society
publication_identifier:
eissn:
- 1460-244X
issn:
- 0024-6115
publication_status: published
publisher: Wiley
quality_controlled: '1'
related_material:
record:
- id: '21050'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Counting equilibria of the electrostatic potential
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 132
year: '2026'
...
---
OA_place: repository
_id: '21050'
abstract:
- lang: eng
text: "In 1873, James C. Maxwell conjectured that the electric field generated by
$n$ point charges in generic position has at most $(n-1)^2$ isolated zeroes. The
first (non-optimal) upper bound was only obtained in 2007 by Gabrielov, Novikov
and Shapiro, who also posed two additional interesting conjectures.\r\n In this
article, we give the best upper bound known to date on the number of zeroes of
the electric field, and construct a counterexample to a conjecture of Gabrielov,
Novikov and Shapiro that the number of equilibria cannot exceed those of the distance
function defined by the unit point charges.\r\n Finally, we note that it is quite
possible that Maxwell's quadratic upper bound is not tight, so it is prudent to
find smaller bounds. Hence, we also explore examples and construct configurations
of charges achieving the highest ratios of the number of electric field zeroes
by point charges found to this day."
article_processing_charge: No
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Gonçalo
full_name: Olivera, Gonçalo
last_name: Olivera
citation:
ama: Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic
potential. arXiv. doi:10.48550/ARXIV.2501.05315
apa: Edelsbrunner, H., Fillmore, C. D., & Olivera, G. (n.d.). Counting equilibria
of the electrostatic potential. arXiv. https://doi.org/10.48550/ARXIV.2501.05315
chicago: Edelsbrunner, Herbert, Christopher D Fillmore, and Gonçalo Olivera. “Counting
Equilibria of the Electrostatic Potential.” ArXiv, n.d. https://doi.org/10.48550/ARXIV.2501.05315.
ieee: H. Edelsbrunner, C. D. Fillmore, and G. Olivera, “Counting equilibria of the
electrostatic potential,” arXiv. .
ista: Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic
potential. arXiv, 10.48550/ARXIV.2501.05315.
mla: Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.”
ArXiv, doi:10.48550/ARXIV.2501.05315.
short: H. Edelsbrunner, C.D. Fillmore, G. Olivera, ArXiv (n.d.).
corr_author: '1'
date_created: 2026-01-27T14:29:27Z
date_published: 2025-03-20T00:00:00Z
date_updated: 2026-06-02T09:24:17Z
day: '20'
department:
- _id: HeEd
doi: 10.48550/ARXIV.2501.05315
external_id:
arxiv:
- '2501.05315'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2501.05315
month: '03'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
record:
- id: '21021'
relation: dissertation_contains
status: public
- id: '21931'
relation: later_version
status: public
status: public
title: Counting equilibria of the electrostatic potential
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: preprint
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20260'
abstract:
- lang: eng
text: 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.
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.\r\nOpen 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."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Erin Wolf
full_name: Chambers, Erin Wolf
last_name: Chambers
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Burning or collapsing
the medial axis is unstable. La Matematica. 2025;4:811-828. doi:10.1007/s44007-025-00170-0
apa: 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
chicago: 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.
ieee: E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Burning
or collapsing the medial axis is unstable,” La Matematica, vol. 4. Springer
Nature, pp. 811–828, 2025.
ista: Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing
the medial axis is unstable. La Matematica. 4, 811–828.
mla: Chambers, Erin Wolf, et al. “Burning or Collapsing the Medial Axis Is Unstable.”
La Matematica, vol. 4, Springer Nature, 2025, pp. 811–28, doi:10.1007/s44007-025-00170-0.
short: E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, La Matematica
4 (2025) 811–828.
corr_author: '1'
date_created: 2025-08-31T22:01:33Z
date_published: 2025-12-01T00:00:00Z
date_updated: 2026-04-07T11:42:48Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s44007-025-00170-0
ec_funded: 1
file:
- access_level: open_access
checksum: e2043259194bfcdf3d74c4da8a5a853f
content_type: application/pdf
creator: dernst
date_created: 2025-12-30T07:52:58Z
date_updated: 2025-12-30T07:52:58Z
file_id: '20885'
file_name: 2025_LaMatematica_Chambers.pdf
file_size: 2678640
relation: main_file
success: 1
file_date_updated: 2025-12-30T07:52:58Z
has_accepted_license: '1'
intvolume: ' 4'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 811-828
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: La Matematica
publication_identifier:
eissn:
- 2730-9657
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '21021'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Burning or collapsing the medial axis is unstable
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 4
year: '2025'
...
---
_id: '17170'
abstract:
- lang: eng
text: "In this article we extend and strengthen the seminal work by Niyogi, Smale,
and Weinberger on the learning of the homotopy type from a sample of an underlying
space. In their work, Niyogi, Smale, and Weinberger studied samples of C² manifolds
with positive reach embedded in ℝ^d. We extend their results in the following
ways: - As the ambient space we consider both ℝ^d and Riemannian manifolds with
lower bounded sectional curvature. - In both types of ambient spaces, we study
sets of positive reach - a significantly more general setting than C² manifolds
- as well as general manifolds of positive reach. - The sample P of a set (or
a manifold) \U0001D4AE of positive reach may be noisy. We work with two one-sided
Hausdorff distances - ε and δ - between P and \U0001D4AE. We provide tight bounds
in terms of ε and δ, that guarantee that there exists a parameter r such that
the union of balls of radius r centred at the sample P deformation-retracts to
\U0001D4AE. We exhibit their tightness by an explicit construction. We carefully
distinguish the roles of δ and ε. This is not only essential to achieve tight
bounds, but also sensible in practical situations, since it allows one to adapt
the bound according to sample density and the amount of noise present in the sample
separately."
acknowledgement: "This research has been supported by the European Research Council
(ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF),
grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant No. I 02979-N35.\r\nWintraecken, Mathijs: 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) grant
No. M-3073, and the welcome package from IDEX of the Université Côte d'Azur."
alternative_title:
- LIPIcs
article_processing_charge: No
arxiv: 1
author:
- first_name: Dominique
full_name: Attali, Dominique
last_name: Attali
- first_name: Hana
full_name: Kourimska, Hana
id: D9B8E14C-3C26-11EA-98F5-1F833DDC885E
last_name: Kourimska
orcid: 0000-0001-7841-0091
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Ishika
full_name: Ghosh, Ishika
id: ee449b28-344d-11ef-a6d5-9ca430e9e9ff
last_name: Ghosh
- first_name: André
full_name: Lieutier, André
last_name: Lieutier
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: 'Attali D, Kourimska H, Fillmore CD, et al. Tight bounds for the learning of
homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and
of Riemannian manifolds. In: 40th International Symposium on Computational
Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024:11:1-11:19.
doi:10.4230/LIPIcs.SoCG.2024.11'
apa: 'Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson,
E. R., & Wintraecken, M. (2024). Tight bounds for the learning of homotopy
à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian
manifolds. In 40th International Symposium on Computational Geometry (Vol.
293, p. 11:1-11:19). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.SoCG.2024.11'
chicago: Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh,
André Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “Tight Bounds
for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of
Euclidean Spaces and of Riemannian Manifolds.” In 40th International Symposium
on Computational Geometry, 293:11:1-11:19. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.11.
ieee: D. Attali et al., “Tight bounds for the learning of homotopy à la Niyogi,
Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds,”
in 40th International Symposium on Computational Geometry, Athens, Greece,
2024, vol. 293, p. 11:1-11:19.
ista: 'Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken
M. 2024. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger
for subsets of euclidean spaces and of Riemannian manifolds. 40th International
Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry,
LIPIcs, vol. 293, 11:1-11:19.'
mla: Attali, Dominique, et al. “Tight Bounds for the Learning of Homotopy à La Niyogi,
Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.”
40th International Symposium on Computational Geometry, vol. 293, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19, doi:10.4230/LIPIcs.SoCG.2024.11.
short: D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson,
M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19.
conference:
end_date: 2024-06-14
location: Athens, Greece
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2024-06-11
date_created: 2024-06-25T11:45:58Z
date_published: 2024-06-06T00:00:00Z
date_updated: 2025-04-15T07:16:57Z
day: '06'
ddc:
- '516'
department:
- _id: GradSch
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.11
ec_funded: 1
external_id:
arxiv:
- '2206.10485'
file:
- access_level: open_access
checksum: 6a2ddc8b51aa58f197a8b294750f1f8d
content_type: application/pdf
creator: cfillmor
date_created: 2024-06-25T11:47:26Z
date_updated: 2024-06-25T11:47:26Z
file_id: '17171'
file_name: LIPIcs.SoCG.2024.11.pdf
file_size: 20886142
relation: main_file
success: 1
file_date_updated: 2024-06-25T11:47:26Z
has_accepted_license: '1'
intvolume: ' 293'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 11:1-11:19
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: 40th International Symposium on Computational Geometry
publication_identifier:
eissn:
- 1868-8969
isbn:
- '9783959773164'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
scopus_import: '1'
status: public
title: Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger
for subsets of euclidean spaces and of Riemannian manifolds
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 293
year: '2024'
...
---
_id: '18097'
abstract:
- lang: eng
text: "In our companion paper \"Tight bounds for the learning of homotopy à la Niyogi,
Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds\"
we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on
a sample P of an input shape \U0001D4AE (either manifold or general set with positive
reach) such that one can infer the homotopy of \U0001D4AE from the union of balls
with some radius centred at P, both in Euclidean space and in a Riemannian manifold
of bounded curvature. The construction showing the optimality of the bounds is
not straightforward. The purpose of this video is to visualize and thus elucidate
said construction in the Euclidean setting."
acknowledgement: "This research has been supported by the European Research Council
(ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF),
grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant No. I02979-N35. Mathijs Wintraecken: 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) grant
No. M-3073, and the welcome package from IDEX of the Université Côte d’Azur.\r\nWe
thank Jean-Daniel Boissonnat, Herbert Edelsbrunner, and Mariette Yvinec for discussion."
alternative_title:
- LIPIcs
article_number: '87'
article_processing_charge: Yes
author:
- first_name: Dominique
full_name: Attali, Dominique
last_name: Attali
- first_name: Hana
full_name: Kourimska, Hana
id: D9B8E14C-3C26-11EA-98F5-1F833DDC885E
last_name: Kourimska
orcid: 0000-0001-7841-0091
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Ishika
full_name: Ghosh, Ishika
id: ee449b28-344d-11ef-a6d5-9ca430e9e9ff
last_name: Ghosh
- first_name: Andre
full_name: Lieutier, Andre
last_name: Lieutier
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: 'Attali D, Kourimska H, Fillmore CD, et al. The ultimate frontier: An optimality
construction for homotopy inference (media exposition). In: 40th International
Symposium on Computational Geometry. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik; 2024. doi:10.4230/LIPIcs.SoCG.2024.87'
apa: 'Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson,
E. R., & Wintraecken, M. (2024). The ultimate frontier: An optimality construction
for homotopy inference (media exposition). In 40th International Symposium
on Computational Geometry (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum
für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2024.87'
chicago: 'Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh,
Andre Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “The Ultimate
Frontier: An Optimality Construction for Homotopy Inference (Media Exposition).”
In 40th International Symposium on Computational Geometry, Vol. 293. Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024. https://doi.org/10.4230/LIPIcs.SoCG.2024.87.'
ieee: 'D. Attali et al., “The ultimate frontier: An optimality construction
for homotopy inference (media exposition),” in 40th International Symposium
on Computational Geometry, Athens, Greece, 2024, vol. 293.'
ista: 'Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken
M. 2024. The ultimate frontier: An optimality construction for homotopy inference
(media exposition). 40th International Symposium on Computational Geometry. SoCG:
Symposium on Computational Geometry, LIPIcs, vol. 293, 87.'
mla: 'Attali, Dominique, et al. “The Ultimate Frontier: An Optimality Construction
for Homotopy Inference (Media Exposition).” 40th International Symposium on
Computational Geometry, vol. 293, 87, Schloss Dagstuhl - Leibniz-Zentrum für
Informatik, 2024, doi:10.4230/LIPIcs.SoCG.2024.87.'
short: D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson,
M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2024.
conference:
end_date: 2024-06-14
location: Athens, Greece
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2024-06-11
corr_author: '1'
date_created: 2024-09-19T10:29:48Z
date_published: 2024-06-06T00:00:00Z
date_updated: 2025-04-15T07:16:58Z
day: '06'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.4230/LIPIcs.SoCG.2024.87
ec_funded: 1
file:
- access_level: open_access
checksum: 9355c2e60b8ec285e1b22719c5b73f1a
content_type: application/pdf
creator: dernst
date_created: 2024-09-19T10:30:37Z
date_updated: 2024-09-19T10:30:37Z
file_id: '18098'
file_name: 2024_LIPICs_Attali.pdf
file_size: 3507177
relation: main_file
success: 1
file_date_updated: 2024-09-19T10:30:37Z
has_accepted_license: '1'
intvolume: ' 293'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: 40th International Symposium on Computational Geometry
publication_identifier:
isbn:
- '9783959773164'
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
quality_controlled: '1'
status: public
title: 'The ultimate frontier: An optimality construction for homotopy inference (media
exposition)'
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 293
year: '2024'
...
---
_id: '11428'
abstract:
- lang: eng
text: The medial axis of a set consists of the points in the ambient space without
a unique closest point on the original set. Since its introduction, the medial
axis has been used extensively in many applications as a method of computing a
topologically equivalent skeleton. 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. 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, demonstrating an isotopy of a shape where the medial axis goes from
collapsible to non-collapsible.
acknowledgement: '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”. Erin Chambers: Supported
in part by the National Science Foundation through grants DBI-1759807, CCF-1907612,
and CCF-2106672. Mathijs Wintraecken: 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 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.'
article_processing_charge: No
author:
- first_name: Erin
full_name: Chambers, Erin
last_name: Chambers
- first_name: Christopher D
full_name: Fillmore, Christopher D
id: 35638A5C-AAC7-11E9-B0BF-5503E6697425
last_name: Fillmore
- first_name: Elizabeth R
full_name: Stephenson, Elizabeth R
id: 2D04F932-F248-11E8-B48F-1D18A9856A87
last_name: Stephenson
orcid: 0000-0002-6862-208X
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: 'Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. A cautionary tale:
Burning the medial axis is unstable. In: Goaoc X, Kerber M, eds. 38th International
Symposium on Computational Geometry. Vol 224. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik; 2022:66:1-66:9. doi:10.4230/LIPIcs.SoCG.2022.66'
apa: 'Chambers, E., Fillmore, C. D., Stephenson, E. R., & Wintraecken, M. (2022).
A cautionary tale: Burning the medial axis is unstable. In X. Goaoc & M. Kerber
(Eds.), 38th International Symposium on Computational Geometry (Vol. 224,
p. 66:1-66:9). Berlin, Germany: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.SoCG.2022.66'
chicago: 'Chambers, Erin, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs
Wintraecken. “A Cautionary Tale: Burning the Medial Axis Is Unstable.” In 38th
International Symposium on Computational Geometry, edited by Xavier Goaoc
and Michael Kerber, 224:66:1-66:9. LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2022. https://doi.org/10.4230/LIPIcs.SoCG.2022.66.'
ieee: 'E. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “A cautionary
tale: Burning the medial axis is unstable,” in 38th International Symposium
on Computational Geometry, Berlin, Germany, 2022, vol. 224, p. 66:1-66:9.'
ista: 'Chambers E, Fillmore CD, Stephenson ER, Wintraecken M. 2022. A cautionary
tale: Burning the medial axis is unstable. 38th International Symposium on Computational
Geometry. SoCG: Symposium on Computational GeometryLIPIcs vol. 224, 66:1-66:9.'
mla: 'Chambers, Erin, et al. “A Cautionary Tale: Burning the Medial Axis Is Unstable.”
38th International Symposium on Computational Geometry, edited by Xavier
Goaoc and Michael Kerber, vol. 224, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2022, p. 66:1-66:9, doi:10.4230/LIPIcs.SoCG.2022.66.'
short: E. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, in:, X. Goaoc,
M. Kerber (Eds.), 38th International Symposium on Computational Geometry, Schloss
Dagstuhl - Leibniz-Zentrum für Informatik, 2022, p. 66:1-66:9.
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