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.
Non-self-intersection is both a topological and a geometric property. It is known that non-self-intersecting regular Bezier curves have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within ℝ3 for a non-self-intersecting, regular C 2 cubic Bezier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper. 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.