Topological and Isotopic Equivalence with Applications to Visualization

There is contemporary interest to preserve appropriate topological characteristics during geometric modeling. Here we focus upon topological equivalence (by home- omorphism) and isotopic equivalence (by ambient isotopy). Homeomorphism is an equivalence relation used for static images, while ambient isotopy requires a homeomorphism at each value of the time parameter, which is particularly applicable for time varying models. We provide sufficient conditions that guarantee these equivalences for geometric approximations of Bezier curves, as one of the fundamental computational representations. We also present further generalization beyond any curve approximation algorithm to establish broad criteria for a sequence of piecewise linear curves to become ambient isotopic to a given smooth curve. The criteria rely upon distance and total curvature, with upper bounds provided. Apart from the major theorems and algorithms, we investigate topological differences based on knot visualization and numerical analysis