When approximating a space curve, it is natural to consider whether the knot type of the original curve is preserved in the approximant. This preservation is of strong contemporary interest in computer graphics and visualization. We establish a criterion to preserve knot type under approximation that relies upon pointwise convergence and convergence in total curvature.
There exist results in the literature showing ambient isotopy from a different point of view @cite_17 @cite_18 . Precisely, there is an upper bound on distance and an upper bound on angles between corresponding points for two curves. If the corresponding distances and angles are within the upper bounds, then they are ambient isotopic. Milnor @cite_15 defined the total curvature for a @math curve using inscribed PL curves. The extension of the definition to piecewise @math curves can be trivially done. Consequently, Fenchel's Theorem can be applied to piecewise @math curves, as we need here.
@cite_18 @cite_15 @cite_17
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It is of increasing contemporary interest to preserve ambient isotopy during geometric modeling. Bezier curves are pervasive in computer aided geometric design, as one of the fundamental computational representations for geometric modeling. For Bezier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge under Hausdorff distance. A natural question is whether subdivision produces topologically reliable PL approximations. Here we focus upon ambient isotopy and prove that sufficiently many subdivisions produce a control polygon ambient isotopic to the Bezier curve. We also derive closed-form formulas to compute the number of subdivision iterations to ensure ambient isotopic equivalence in the resulting approximation. This work relies upon explicitly constructing homeomorphism and ambient isotopy, which provides more algorithmic efficiency than only showing the existence of these equivalence relations. We define a new notion of total curvature, called net total curvature, for finite graphs embedded inR n , and investigate its properties. Two guiding principles ar e given by Milnor's way of measuring the local crookedness of a Jorda n curve via a Crofton-type formula, and by considering the double cover of a given graph as an Eulerian circuit. The strength of combining these ideas in defining the curvature f unctional is (1) it allows us to interpret the singular non-eulidean behavior at the vertices of the graph as a superposition of vertices of a 1-dimensional manifold, and thus (2) one can compute the total curva- ture for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the cur vature. A collection of results on upper lower bounds of the total curvature on isotopy homeomorphism classes of embed- dings is presented, which in turn demonstrates the effectiveness of net total curvature as a new functional measuring complexity of spatial graphs in differential-geometric terms. Generalizing Milnor’s result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result for essential subarcs of a knot. Ambient isotopic approximations are fundamental for correct representation of the embedding of geometric objects in R 3, with a detailed geometric construction given here. Using that geometry, an algorithm is presented for efficient update of these isotopic approximations for dynamic visualization with a molecular simulation.