Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Recovery problem of parametrizations from Legendre data

C. Muñoz-Cabello, T. Nishimura, R. Oset Sinha

Abstract

The problem of recovery of parametrizations from Legendre data is a very important inverse problem. In this paper, we provide a systematic and widely-applicable method to recover parametrizations $f: U_n \to \mathbb{R}^{n+1}$ from Legendre data where $U_n$ is an open subset of $\mathbb R^n$. Namely, for a dense subset of the space of real-analytic parametrizations from $U_n$ into $\mathbb{R}^{n+1}$, we show how to recover the parametrization from the Gauss mapping and the height function. Moreover, in order to assist readers to apply results of this paper, many concrete examples are given.

Recovery problem of parametrizations from Legendre data

Abstract

The problem of recovery of parametrizations from Legendre data is a very important inverse problem. In this paper, we provide a systematic and widely-applicable method to recover parametrizations from Legendre data where is an open subset of . Namely, for a dense subset of the space of real-analytic parametrizations from into , we show how to recover the parametrization from the Gauss mapping and the height function. Moreover, in order to assist readers to apply results of this paper, many concrete examples are given.
Paper Structure (8 sections, 7 theorems, 108 equations, 2 figures)

This paper contains 8 sections, 7 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem 1
  4. Proof of Theorem \ref{['theorem2']}
  5. Proof of the assertion (1) of Theorem \ref{['theorem2']}
  6. Proof of the assertion (2) of Theorem \ref{['theorem2']}
  7. Proof of Theorem 3
  8. Examples

Key Result

Theorem 1

Let $f: U_n\to \mathbb{R}^{n+1}$ be a regular frontal. Then, $f$ is recovered from its Legendre data $\left\{\nu({\color{black}\bf x}), a({\color{black}\bf x})\right\}_{{\color{black}\bf x}\in Reg({\color{black}\nu})}$. Namely, for any ${\color{black}\bf x}\in U_n$, $f({\color{black}\bf x})$ can be

Figures (2)

  • Figure 1: Images of the mappings from Example \ref{['example1']}, in order of appearance.
  • Figure 2: Image of the $D_4^+$ (top) and $D_4^-$ (bottom) singularities, projected into $\mathbb{R}^3$ via the map $(u,x,y,z) \mapsto (x,y,z)$ for different values of $u$.

Theorems & Definitions (25)

  • Definition 1
  • Example 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 2
  • Definition 5
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • ...and 15 more