Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hedgehog Reconstruction of Polygons: Non-Central Sections and Slabs

Brendan Matthews

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

Hedgehog Reconstruction of Polygons: Non-Central Sections and Slabs

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
Paper Structure (10 sections, 6 theorems, 45 equations, 4 figures)

This paper contains 10 sections, 6 theorems, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Support function and envelopes
  4. Hedgehogs
  5. Real Analytic Functions
  6. Proofs of the Theorems
  7. Proof of Theorem \ref{['Thm1']}
  8. Case l1 perp = m1 perp (l1 is parallel to m1)
  9. Case l1 perp = m2 perp (l1 is parallel to m2)
  10. Proof of Theorem \ref{['Thm2']}

Key Result

Theorem 1.1

Let $P$ and $Q$ be convex polygons in $\mathbb{R}^2$ with a non-trivial piecewise analytic hedgehog $M$ in their interiors. If for every supporting line $H$ of $M$, then $P= Q$.

Figures (4)

  • Figure 1: $P \neq Q$, but $\text{vol}_{2}\left( P \cap S_M(\xi) \right) = \text{vol}_{2}\left( Q \cap S_M(\xi) \right)$ for all $\xi \in S^1$.
  • Figure 2: Convex and Non-Convex Bodies in $\mathbb{R}^2$.
  • Figure 3: Intersection of a slab $S_E(\xi)$ with $P$.
  • Figure 4: Analytic hedgehogs.

Theorems & Definitions (13)

  • Conjecture 1
  • Theorem 1.1
  • Conjecture 2
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3: KrantzParks2002, p. 3
  • Theorem 2.1: KrantzParks2002, pp. 4--6
  • Theorem 2.2: KrantzParks2002, p. 13
  • ...and 3 more