Topology during Subdivision of Bézier Curves II: Ambient Isotopy

It is of increasing contemporary interest to preserve ambient isotopy during geometric modeling. Bézier curves are pervasive in computer aided geometric design, as one of the fundamental computational representations for geometric modeling. For Bézier 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 Bézier 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.