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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 computer-aided geometric modeling. For practical geometric modeling algorithms, it is often desirable to create piecewise linear approximations to compact manifolds embedded in R ³, 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 R ³, there exists a piecewise linear ambient isotopic approximation. Furthermore, this isotopy has compact support, with specific bounds upon the size …