This paper describes an algorithm for computing all irreducible components of a (not necessarily reduced) algebraic curve \(\mathcal C\) defined over \(\mathbb Q\) and embedded in a space of arbitrary dimension. The classical procedure is to take a generic projection of the curve to a plane and factorize the equation of the plane projection. The difficulty with this is that the complexity of projection increases drastically with the degree of the curve, so that the problem may become intractable for a curve with many components, even if each component has small degree. The approach of the present paper is to mitigate this difficulty by working in a plane section and using floating-point arithmetic when possible. The following outline of the algorithm is taken from the introduction: ``First, we compute the points of a generic plane section of the considered curve \(\mathcal C\). Then we describe an infinitesimal neighbourhood of this section (i.e. a ``fat'' section) by computing approximate Taylor expansions along the curve. Using a combinatorial procedure (zero-sums search) we compute a partition of these points. This partition is indeed the partition induced by the hyper-plane sections of the irreducible components of \(\mathcal C\). Then using Newton-Hensel liftings we compute numerical approximations of the factors of the equation of a generic projection of \(\mathcal C\). Finally we construct, by rational approximation, an exact candidate for the equation of a generic projection of \(\mathcal C\) and a candidate for its absolute factorization and the algebraic extension of \(\mathbb Q\). Then we check the exact validity of these representations of the irreducible branches''.