Hedgehog Reconstruction of Polygons: Non-Central Sections and Slabs

TL;DR

This paper demonstrates that polygons can be uniquely reconstructed from the lengths of non-central sections and slab areas supporting specific hedgehog functions, advancing geometric reconstruction techniques.

Contribution

It introduces novel methods for polygon reconstruction using non-central section lengths and slab areas linked to analytic hedgehog functions.

Findings

Unique reconstruction of polygons from non-central section lengths

Reconstruction based on slab areas for centrally-symmetric polygons

Extension of hedgehog-based geometric analysis

Abstract

We show that a polygon can be uniquely determined by the lengths of non-central sections supporting a piecewise-analytic hedgehog in the interior of the polygon. We also prove the analogous result for slab areas - centrally-symmetric polygon can be reconstructed based on the areas of slabs supporting an analytic centrally-symmetric hedgehog in the interior of the polygon