Adiabatic isometric mapping algorithm for embedding 2-surfaces in Euclidean 3-space

Shannon Ray Warner A Miller Paul M Alsing Shing-Tung Yau

Mathematical Physics mathscidoc:1912.43703

Classical and Quantum Gravity, 32, (23), 235012, 2015.11
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 on a given polyhedral metric to produce an embedding that approximates the unique Alexandrov polyhedron. 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
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@inproceedings{shannon2015adiabatic,
  title={Adiabatic isometric mapping algorithm for embedding 2-surfaces in Euclidean 3-space},
  author={Shannon Ray, Warner A Miller, Paul M Alsing, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205109123707267},
  booktitle={Classical and Quantum Gravity},
  volume={32},
  number={23},
  pages={235012},
  year={2015},
}
Shannon Ray, Warner A Miller, Paul M Alsing, and Shing-Tung Yau. Adiabatic isometric mapping algorithm for embedding 2-surfaces in Euclidean 3-space. 2015. Vol. 32. In Classical and Quantum Gravity. pp.235012. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224205109123707267.
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