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International audience; PTOPO is a MAPLE package computing the topology and describing the geometry of a parametric plane curve. The algorithm behind PTOPO constructs an abstract graph that is isotopic to the curve. PTOPO exploits the benefits of the parametric representation and performs all computations in the parameter space using exact computing. PTOPO computes the topology and visualizes the curve in less than a second for most examples in the literature.

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 OB (nd6 + nd5 τ + d4 (n2 + nτ) + d3 (n2τ + n3) + n3d2τ). This bound matches the current record bound OB (d6 + d5 τ) 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 OB (N6), which improves the state-of-the-art result, due to Alcazar and Diaz-Toca [CAGD'10], by a factor of N10. However, visualizing the curve on top of the abstract graph construction, increases the bound to OB (N7). We have implemented PTOPO in maple for the case of planar curves. Our experiments illustrate its practical nature.