Computational topology for isotopic surface reconstruction

Authors:
K. Abe;J. Bisceglio;D. R. Ferguson;T. J. Peters;A. C. Russell;T. Sakkalis
Affiliations:
Department of Mathematics, University of Connecticut, Storrs, CT;Department of Computer Science, University of Connecticut, Storrs, CT;DRF Associates, Seattle, WA;Department of Computer Science and Engineering and Department of Mathematics, University of Connecticut, Storrs, CT;Department of Computer Science and Engineering, University of Connecticut, Storrs, CT;Agricultural University of Athens, Athens, Greece and Massachusetts Institute of Technology, Cambridge, MA
Venue:
Theoretical Computer Science - Spatial representation: Discrete vs. continous computational models
Year:
2006

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Abstract

New computational topology techniques are presented for surface reconstruction of 2-manifolds with boundary, while rigorous proofs have previously been limited to surfaces without boundary. This is done by an intermediate construction of the envelope (as defined herein) of the original surface. For any compact C2-manifold M embedded in R3, it is shown that its envelope is C1,1. Then it is shown that there exists a piecewise linear (PL) subset of the reconstruction of the envelope that is ambient isotopic to M, whenever M is orientable. The emphasis of this paper is upon the formal mathematical proofs needed for these extensions. (Practical application examples have already been published in a companion paper.) Possible extensions to non-orientable manifolds are also discussed. The mathematical exposition relies heavily on known techniques from differential geometry and topology, but the specific new proofs are intended to be sufficiently specialized to prompt further algorithmic discoveries.