Recovery problem of parametrizations from Legendre data

TL;DR

This paper introduces a systematic method to recover parametrizations from Legendre data, crucial for inverse problems, by utilizing Gauss mapping and height functions, with numerous concrete examples provided.

Contribution

The paper presents a widely applicable technique for recovering parametrizations from Legendre data, expanding the toolkit for inverse geometric problems.

Findings

Method works for a dense subset of real-analytic parametrizations

Recovery uses Gauss mapping and height function

Numerous concrete examples demonstrate applicability

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.