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The medial axis of a smoothly embedded surface in \({{\mathbb R}}^3\) consists of all points for which the Euclidean distance function on the surface has at least two global minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided.

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The Mid-sphere Cousin of the Medial Axis Transform

  • 会议论文

  • 发表日期：

    2026年

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  • Herbert Edelsbrunner,
  • Elizabeth Stephenson,
  • Martin Hafskjold Thoresen

摘要

The medial axis of a smoothly embedded surface in \({{\mathbb R}}^3\) consists of all points for which the Euclidean distance function on the surface has at least two global minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided.