Approximating the Nonlocal Curvature of Planar Curves
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916685481508864 |
|---|---|
| author | Fleming, Cole Seguin, Brian |
| author_facet | Fleming, Cole Seguin, Brian |
| contents | Here we establish several results on the nonlocal curvature of planar curves. First we show how to express the nonlocal curvature of a curve relative to a point in terms of the nonlocal curvatures of simpler components of that curve relative to the same point. To obtain these results, it is necessary to extend the definition of nonlocal curvature to points off of the curve. We also find a formula for the nonlocal curvature of a line segment relative to any point in the plane in terms of the incomplete beta function. These results are then used to prove an approximation theorem, which states that the nonlocal curvature of a planar curve with some Hölder regularity can be approximated by the nonlocal curvature of a linear interpolating spline associated with the curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximating the Nonlocal Curvature of Planar Curves Fleming, Cole Seguin, Brian Differential Geometry 65D15 (Primary) 49Q05 (Secondary) Here we establish several results on the nonlocal curvature of planar curves. First we show how to express the nonlocal curvature of a curve relative to a point in terms of the nonlocal curvatures of simpler components of that curve relative to the same point. To obtain these results, it is necessary to extend the definition of nonlocal curvature to points off of the curve. We also find a formula for the nonlocal curvature of a line segment relative to any point in the plane in terms of the incomplete beta function. These results are then used to prove an approximation theorem, which states that the nonlocal curvature of a planar curve with some Hölder regularity can be approximated by the nonlocal curvature of a linear interpolating spline associated with the curve. |
| title | Approximating the Nonlocal Curvature of Planar Curves |
| topic | Differential Geometry 65D15 (Primary) 49Q05 (Secondary) |
| url | https://arxiv.org/abs/2504.08168 |