@article{12287,
  abstract     = {We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.},
  author       = {Boissonnat, Jean-Daniel and Dyer, Ramsay and Ghosh, Arijit and Wintraecken, Mathijs},
  issn         = {1432-0444},
  journal      = {Discrete & Computational Geometry},
  keywords     = {Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Theoretical Computer Science},
  pages        = {156--191},
  publisher    = {Springer Nature},
  title        = {{Local criteria for triangulating general manifolds}},
  doi          = {10.1007/s00454-022-00431-7},
  volume       = {69},
  year         = {2023},
}

@article{12763,
  abstract     = {Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift 176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). Bangert showed that sets of positive reach in Euclidean space and Riemannian manifolds are very similar. In this paper we introduce a slight variant of Kleinjohann’s and Bangert’s extension and quantify the similarity between sets of positive reach in Euclidean space and Riemannian manifolds in a new way: Given p∈M and q∈S, we bound the local feature size (a local version of the reach) of its lifting to the tangent space via the inverse exponential map (exp−1p(S)) at q, assuming that rchM(S) and the geodesic distance dM(p,q) are bounded. These bounds are motivated by the importance of the reach and local feature size to manifold learning, topological inference, and triangulating manifolds and the fact that intrinsic approaches circumvent the curse of dimensionality.},
  author       = {Boissonnat, Jean Daniel and Wintraecken, Mathijs},
  issn         = {2367-1734},
  journal      = {Journal of Applied and Computational Topology},
  pages        = {619--641},
  publisher    = {Springer Nature},
  title        = {{The reach of subsets of manifolds}},
  doi          = {10.1007/s41468-023-00116-x},
  volume       = {7},
  year         = {2023},
}

@article{12960,
  abstract     = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e., submanifolds of Rd defined as the zero set of some multivariate multivalued smooth function f:Rd→Rd−n, where n is the intrinsic dimension of the manifold. A natural way to approximate a smooth isomanifold M=f−1(0) is to consider its piecewise linear (PL) approximation M^
 based on a triangulation T of the ambient space Rd. In this paper, we describe a simple algorithm to trace isomanifolds from a given starting point. The algorithm works for arbitrary dimensions n and d, and any precision D. Our main result is that, when f (or M) has bounded complexity, the complexity of the algorithm is polynomial in d and δ=1/D (and unavoidably exponential in n). Since it is known that for δ=Ω(d2.5), M^ is O(D2)-close and isotopic to M
, our algorithm produces a faithful PL-approximation of isomanifolds of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality reduction techniques, the dependency on d in the size of M^ can be completely removed with high probability. We also show that the algorithm can handle isomanifolds with boundary and, more generally, isostratifolds. The algorithm for isomanifolds with boundary has been implemented and experimental results are reported, showing that it is practical and can handle cases that are far ahead of the state-of-the-art. },
  author       = {Boissonnat, Jean Daniel and Kachanovich, Siargey and Wintraecken, Mathijs},
  issn         = {1095-7111},
  journal      = {SIAM Journal on Computing},
  number       = {2},
  pages        = {452--486},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Tracing isomanifolds in Rd in time polynomial in d using Coxeter–Freudenthal–Kuhn triangulations}},
  doi          = {10.1137/21M1412918},
  volume       = {52},
  year         = {2023},
}

@inproceedings{13048,
  abstract     = {In this paper we introduce a pruning of the medial axis called the (λ,α)-medial axis (axλα). We prove that the (λ,α)-medial axis of a set K is stable in a Gromov-Hausdorff sense under weak assumptions. More formally we prove that if K and K′ are close in the Hausdorff (dH) sense then the (λ,α)-medial axes of K and K′ are close as metric spaces, that is the Gromov-Hausdorff distance (dGH) between the two is 1/4-Hölder in the sense that dGH (axλα(K),axλα(K′)) ≲ dH(K,K′)1/4. The Hausdorff distance between the two medial axes is also bounded, by dH (axλα(K),λα(K′)) ≲ dH(K,K′)1/2. These quantified stability results provide guarantees for practical computations of medial axes from approximations. Moreover, they provide key ingredients for studying the computability of the medial axis in the context of computable analysis.},
  author       = {Lieutier, André and Wintraecken, Mathijs},
  booktitle    = {Proceedings of the 55th Annual ACM Symposium on Theory of Computing},
  isbn         = {9781450399135},
  location     = {Orlando, FL, United States},
  pages        = {1768--1776},
  publisher    = {Association for Computing Machinery},
  title        = {{Hausdorff and Gromov-Hausdorff stable subsets of the medial axis}},
  doi          = {10.1145/3564246.3585113},
  year         = {2023},
}

@inproceedings{11428,
  abstract     = {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.},
  author       = {Chambers, Erin and Fillmore, Christopher D and Stephenson, Elizabeth R and Wintraecken, Mathijs},
  booktitle    = {38th International Symposium on Computational Geometry},
  editor       = {Goaoc, Xavier and Kerber, Michael},
  isbn         = {978-3-95977-227-3},
  issn         = {1868-8969},
  location     = {Berlin, Germany},
  pages        = {66:1--66:9},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{A cautionary tale: Burning the medial axis is unstable}},
  doi          = {10.4230/LIPIcs.SoCG.2022.66},
  volume       = {224},
  year         = {2022},
}