New efficient algorithm for the isometric embedding of 2-surface metrics in 3 dimensional Euclidean space

TL;DR

This paper introduces a new numerical algorithm for efficiently computing the isometric embedding of 2-dimensional surface metrics into 3D Euclidean space, using spectral collocation and Newton-Raphson methods.

Contribution

The paper presents a novel, efficient numerical approach for isometric embedding of 2-surfaces, combining spectral collocation with a Newton-Raphson solver for the fundamental embedding equations.

Findings

Method converges with suitable initial guesses

Efficient for smooth 2-metrics

Detailed algorithm implementation provided

Abstract

We present a new numerical method for the isometric embedding of 2-geometries specified by their 2-metrics in three dimensional Euclidean space. Our approach is to directly solve the fundamental embedding equation supplemented by six conditions that fix translations and rotations of the embedded surface. This set of equations is discretized by means of a pseudospectral collocation point method. The resulting nonlinear system of equations are then solved by a Newton-Raphson scheme. We explain our numerical algorithm in detail. By studying several examples we show that our method converges provided we start the Newton-Raphson scheme from a suitable initial guess. Our novel method is very efficient for smooth 2-metrics.