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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)

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)

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)

Reconstruction of surfaces with boundary remains a challenge. Recent theoretical advances [Abe et al. 2006] define envelopes as surfaces without boundary to approximate those with boundary. Figure 1 shows a Möbius strip on the left, denoted as <i>M</i>. An illustration of its envelope appears next, intuitively understood as attaching a small ball to… (More)

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