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Open Access Publications from the University of California

Computational Topology: Ambient Isotopic Approximation of 2-Manifolds

  • Author(s): Amenta, Nina
  • Peters, Thomas J.
  • Russell, Alexander
  • et al.

A fundamental issue in theoretical computer science is that of establishing unambiguous formal criteria for algorithmic output. This paper does so within the domain of computeraided geometric modeling. For practical geometric modeling algorithms, it is often desirable to create piecewise linear approximations to compact manifolds embedded in R3, and it is usually desirable for these two representations to be topologically equivalent. Though this has traditionally been taken to mean that the two representations are homeomorphic, such a notion of equivalence suffers from a variety of technical and philosophical difficulties; we adopt the stronger notion of ambient isotopy. It is shown here, that for any C2, compact, 2-manifold without boundary, which is embedded in R3, there exists a piecewise linear ambient isotopic approximation. Furthermore, this isotopy has compact support, with specific bounds upon the size of this compact neighborhood. These bounds may be useful for practical application in computer graphics and engineering design simulations. The proof given relies upon properties of the medial axis, which is explained in this paper.

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