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DOI: 10.4230/LIPIcs.SoCG.2026.92

Estimation of Conformal Metrics

Authors: Jérôme Taupin

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

We study deformations of the geodesic distances on a domain of ℝ^N induced by a function called conformal factor. We show that under a positive reach assumption on the domain (not necessarily a submanifold) and mild assumptions on the conformal factor, geodesics for the conformal metric have good regularity properties in the form of a lower bounded reach. This regularity allows for efficient estimation of the conformal metric from a random point cloud with a relative error proportional to the Hausdorff distance between the point cloud and the original domain. We then establish convergence rates of order n^{-1/d} that are close to sharp when the intrinsic dimension d of the domain is large, for an estimator that can be computed in O(n²) time. Finally, this paper includes a useful equivalence result between ball graphs and nearest-neighbors graphs when assuming Ahlfors regularity of the sampling measure, allowing to transpose results from one setting to another.

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Jérôme Taupin. Estimation of Conformal Metrics. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 92:1-92:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{taupin:LIPIcs.SoCG.2026.92,
  author =	{Taupin, J\'{e}r\^{o}me},
  title =	{{Estimation of Conformal Metrics}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{92:1--92:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.92},
  URN =		{urn:nbn:de:0030-drops-258986},
  doi =		{10.4230/LIPIcs.SoCG.2026.92},
  annote =	{Keywords: Geometric inference, metric estimation, conformal metric, geodesics, sets of positive reach}
}

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Estimation of Conformal Metrics

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