research-article

Any open bounded subset of R n has the same homotopy type as its medial axis

Author: André LieutierAuthors Info & Claims

Pages 1029 - 1046

https://doi.org/10.1016/j.cad.2004.01.011

Published: 15 September 2004 Publication History

Abstract

Medial axis Transform is sometimes used as an intermediate representation in algorithms for meshing or recognition of shapes from digitized data. This raises the question whether the Medial Axis captures fundamental topological invariants of the object. The (positive) answer has been known already in the case of smooth objects. The main result presented here is the homotopy equivalence of any bounded open subset of R n with its medial axis.

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Cited By

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Cotsakis R(2025)Computable Bounds for the Reach and r-Convexity of Subsets of Discrete & Computational Geometry10.1007/s00454-023-00624-873:1(92-128)Online publication date: 1-Jan-2025
https://dl.acm.org/doi/10.1007/s00454-023-00624-8
Wang NHuang HSong SWang BWang WGuo X(2024)MATTopo: Topology-preserving Medial Axis Transform with Restricted Power DiagramACM Transactions on Graphics10.1145/368776343:6(1-16)Online publication date: 19-Dec-2024
https://dl.acm.org/doi/10.1145/3687763
Lieutier AWintraecken MSaha BServedio R(2023)Hausdorff and Gromov-Hausdorff Stable Subsets of the Medial AxisProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585113(1768-1776)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585113
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Any open bounded subset of R n has the same homotopy type as its medial axis

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Published In

Computer-Aided Design Volume 36, Issue 11

Sep 2004

106 pages

ISSN:0010-4485

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Elsevier Ltd.

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Butterworth-Heinemann

United States

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Published: 15 September 2004

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Cotsakis R(2025)Computable Bounds for the Reach and r-Convexity of Subsets of Discrete & Computational Geometry10.1007/s00454-023-00624-873:1(92-128)Online publication date: 1-Jan-2025
https://dl.acm.org/doi/10.1007/s00454-023-00624-8
Wang NHuang HSong SWang BWang WGuo X(2024)MATTopo: Topology-preserving Medial Axis Transform with Restricted Power DiagramACM Transactions on Graphics10.1145/368776343:6(1-16)Online publication date: 19-Dec-2024
https://dl.acm.org/doi/10.1145/3687763
Lieutier AWintraecken MSaha BServedio R(2023)Hausdorff and Gromov-Hausdorff Stable Subsets of the Medial AxisProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585113(1768-1776)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585113
Sivignon I(2022)Exact and Optimal Conversion of a Hole-free 2d Digital Object into a Union of Balls in Polynomial TimeDiscrete Geometry and Mathematical Morphology10.1007/978-3-031-19897-7_30(382-394)Online publication date: 24-Oct-2022
https://dl.acm.org/doi/10.1007/978-3-031-19897-7_30
Hornus SKuipers TDevillers OTeillaud MMartínez JGlisse MLazard SLefebvre S(2020)Variable-width contouring for additive manufacturingACM Transactions on Graphics10.1145/3386569.339244839:4(131:1-131:17)Online publication date: 12-Aug-2020
https://dl.acm.org/doi/10.1145/3386569.3392448

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Any open bounded subset of Rn has the same homotopy type than its medial axis

Homotopy-preserving medial axis simplification

Dynamic Skeletonization via Variational Medial Axis Sampling

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Any open bounded subset of Rⁿ has the same homotopy type than its medial axis