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

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.