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Given a nonsingular compact two-manifold F without boundary, we present methods for establishing a family of surfaces which can approximate F so that each approximant is ambient isotopic to F: The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximations, for applications in graphics, animation and surface… (More)

We report new techniques and theory in computational topology for reconstructing surfaces with boundary. This complements and extends known techniques for surfaces without boundary. Our approach is motivated by differential geometry and differential topology. We have also conducted significant experimental work to test our resultant implementations. We… (More)

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 C 2 manifold M embedded in R 3 , it is shown… (More)

- T J Peters, J Bisceglio, D R Ferguson, C M Hoffmann, T Maekawa, N M Patrikalakis +2 others
- 2004

The Boolean algebra of regular closed sets is prominent in topol-ogy, particularly as a dual for the Stone-ˇ Cech compactification. This algebra is also central for the theory of geometric computation, as a representation for combinatorial operations on geometric sets. However, the issue of computational approximation introduces unresolved subtleties that… (More)

For a rich class of composite cubic Bézier curves, an a priori bound exists on the number of subdivisions to achieve ambient isotopy between the curve and its control polygon. The authors of that theorem did not present any examples when the original control polygon is not ambient isotopic to the curve. An example is given here of a composite cubic Bézier… (More)

Given a nonsingular compact 2-manifold ¦ without boundary, we present methods for establishing a family of surfaces which can approximate ¦ so that each approximant is ambient isotopic to ¦. The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximations, for applications in graphics, animation and surface… (More)

This paper presents computational topology techniques for reconstruction of surfaces with boundary, where all manifolds considered are assumed to be embedded in R 3. The focus here is upon examples and applications, with the theoretical basis being presented in a companion paper. As a step towards these results, we consider any C 2 compact 2-manifold M with… (More)

Meshes with T-joints (T-meshes) and related high-order surfaces have many advantages in situations where flexible local refinement is needed. At the same time, designing subdivision rules and bases for T-meshes is much more difficult, and fewer options are available. For common geometric modeling tasks it is desirable to retain the simplicity and… (More)

This paper presents new mathematical foundations for topologically correct surface reconstruction techniques that are applicable to 2-manifolds with boundary, where provable techniques previously had 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… (More)

We use notation introduced in the Appendix of the paper. To compute the value of P 1 j we need to define a stencil of control points P 0 i that influence its value. We exhaustively enumerate the 1-ring neighborhood configurations (with a suitable extension at T-vertices and T-edges) of a vertex in all possible T-mesh configurations. Here we show that such a… (More)