The chordal distance transform of geometric loops and its persistent homology

Binnie, James A. D. ORCID: https://orcid.org/0009-0006-1741-7410, Sumray, Otto and Yim, Ka Man 2026. The chordal distance transform of geometric loops and its persistent homology. Cornell University. 10.48550/arXiv.2603.17125

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Abstract

We present an isometry and parametrisation invariant of embeddings of S1 into Euclidean space. We do so by representing the distance between pairs of points on the embedded circle as a function on a Mobius band, the two-point finite subset space of S1. We call this function the chordal distance transform of the embedding. We show that the sublevel set persistent homology of the chordal distance transform satisfies the desired isometry and parametrisation invariance, and is a continuous transform with respect to the Whitney topology on the space of circle embeddings and the bottleneck distance in the space of persistence diagrams. We then considered the generic behaviour of the chordal distance transform for C2 and finite piecewise linear embeddings. In the C2-case, we show that non-boundary critical points of the chordal distance transform are finite and non-degenerate on an open and dense subset of circle embeddings. Consequently, its persistent homology is pointwise finite dimensional for generic C2-embeddings. In the finite piecewise linear case, we also find piecewise-continuous analogues of non-degenerate critical points, and give generic conditions for the homological critical points of the chordal distance transform to be non-degenerate. In order to gain a geometric interpretation of the chordal distance transform and its persistent homology, we give a geometric characterisation of the C2 and finite piecewise linear non-degenerate critical points. Finally, we consider how the chordal distance transform can be generalised to capture geometric features involving n ≥ 2 points on an embedded shape, as a function on the n-point finite subset space.

Item Type:	Working paper
Date Type:	Submission
Status:	Published
Schools:	Schools > Mathematics
Publisher:	Cornell University
Last Modified:	20 Mar 2026 13:15
URI:	https://orca.cardiff.ac.uk/id/eprint/185885

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