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.
The preservation of topology in computer graphics and visualization has previously been articulated in two primary applications @cite_16 : preservation of isotopic equivalence by approximations; and preservation of isotopic equivalence during dynamic changes, such as protein unfolding.
@cite_16
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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.