Adiabatic Isometric Mapping Algorithm for Embedding 2-Surfaces in Euclidean 3-Space

TL;DR

The paper introduces the AIM algorithm for efficiently embedding 2-surfaces in 3D space, based on Alexandrov's theorem, and demonstrates its effectiveness on various polyhedral metrics, including non-convex cases.

Contribution

It presents a novel, polynomial-time algorithm for isometric embedding of 2-surfaces, extending Alexandrov's theorem to practical computational methods.

Findings

AIM algorithm runs in sub-cubic time relative to vertices.

Successfully embeds non-convex and complex surfaces.

Demonstrates applicability to surfaces near black hole ergospheres.

Abstract

Alexandrov proved that any simplicial complex homeomorphic to a sphere with strictly non-negative Gaussian curvature at each vertex can be isometrically embedded uniquely in $\mathbb{R}^3$ as a convex polyhedron. Due to the nonconstructive nature of his proof, there have yet to be any algorithms, that we know of, that realizes the Alexandrov embedding in polynomial time. Following his proof, we developed the adiabatic isometric mapping (AIM) algorithm. AIM uses a guided adiabatic pull-back procedure to produce "smooth" embeddings. Tests of AIM applied to two different polyhedral metrics suggests that its run time is sub cubic with respect to the number of vertices. Although Alexandrov's theorem specifically addresses the embedding of convex polyhedral metrics, we tested AIM on a broader class of polyhedral metrics that included regions of negative Gaussian curvature. One test was on a surface just outside the ergosphere of a Kerr black hole.