On the geometry and the topology of parametric curves

We consider the problem of computing the topology and describing the geometry of a parametric curve in Rn. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization. Most importantly, we perform all computations in the parameter space and not in the implicit space. When the parametrization involves polynomials of degree at most d and maximum bitsize of coefficients τ, then the worst case bit complexity of PTOPO is ÕB (nd6 + nd5τ + d4 (n2+ nτ) + d3 (n2τ + n3) + n3d2τ). This bound matches the current record bound ÕB (d6 + d5τ) for the problem of computing the topology of a planar algebraic curve given in implicit form. For planar and space curves, if N = max{d, τ}, the complexity of PTOPO becomes ÕB (N6), which improves the state-of-the-art result, due to Alcázar and Díaz-Toca [CAGD'10], by a factor of N10. However, visualizing the curve on top of the abstract graph construction, increases the bound to ÕB (N7). We have implemented PTOPO in maple for the case of planar curves. Our experiments illustrate its practical nature.