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We show how to compute the planar arrangement induced by segments of arbitrary algebraic curves with the Bentley-Ottmann sweep-line algorithm. The necessary geometric primitives reduce to cylindrical algebraic decompositions of the plane for one or two curves. We compute them by a new and efficient method that combines adaptive-precision root finding (the Bitstream Descartes method of Eigenwillig et al., 2005) with a small number of symbolic computations, and that delivers the exact result in all cases. Thus we obtain an algorithm which produces the mathematically true arrangement, undistorted by rounding error, for any set of input segments. Our algorithm is implemented in the EXACUS library AlciX. We report on experiments; they indicate the efficiency of our approach.