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Coarse-grained curvature tensor on polygonal surfaces

by Charlie Duclut, Aboutaleb Amiri, Joris Paijmans, Frank Jülicher

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Submission summary

As Contributors: Aboutaleb Amiri · Charlie Duclut
Arxiv Link: https://arxiv.org/abs/2104.07988v2 (pdf)
Date accepted: 2022-01-03
Date submitted: 2021-11-09 08:22
Submitted by: Duclut, Charlie
Submitted to: SciPost Physics Core
Academic field: Physics
Specialties:
  • Differential Geometry
  • Mathematical Physics
  • Computational Geometry
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational

Abstract

Using concepts from integral geometry, we propose a definition for a local coarse-grained curvature tensor that is well-defined on polygonal surfaces. This coarse-grained curvature tensor shows fast convergence to the curvature tensor of smooth surfaces, capturing with accuracy not only the principal curvatures but also the principal directions of curvature. Thanks to the additivity of the integrated curvature tensor, coarse-graining procedures can be implemented to compute it over arbitrary patches of polygons. When computed for a closed surface, the integrated curvature tensor is identical to a rank-2 Minkowski tensor. We also provide an algorithm to extend an existing C++ package, that can be used to compute efficiently local curvature tensors on triangulated surfaces.

Published as SciPost Phys. Core 5, 011 (2022)



List of changes

- we have modified our statements regarding continuity throughout the manuscript (in the introduction, in the conclusion and in section I);
- we have changed the discussion regarding the weighting fraction throughout the manuscript (in the introduction, in the conclusion and in section I);
- we have included Eqs. (10) and (11) to the main text;
- we have added references to relevant and seminal work in integral geometry.

Submission & Refereeing History

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Resubmission 2104.07988v2 on 9 November 2021

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