Abstract
The problem of computing the topology of curves has received special attention from both Computer Aided Geometric Design and Symbolic Computation. It is well known that the general position condition simplifies the computation of the topology of a real algebraic plane curve defined implicitly since, under this assumption, singular points can be presented in a very convenient way for that purpose. Here we will show how the topology of cubic, quartic and quintic plane curves can be computed in the same manner even if the curve is not in general position, avoiding thus coordinate changes. This will be possible by applying new formulae, derived from subresultants, which describe multiple roots of univariate polynomials as rational functions of the considered polynomial coefficients. We will also characterize those higher degree curves where this approach can be used and use this technique to describe the curve arising when intersecting two ellipsoids.
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Acknowledgements
The authors are partially supported by the grant PID2020-113192GB-I00/AEI/ 10.13039/501100011033 (Mathematical Visualization: Foundations, Algorithms and Applications) from the Spanish Agencia Estatal de Investigación (Ministerio de Ciencia e Innovación). J. Caravantes belongs to the Research Group ASYNACS (Ref. CT-CE2019/683).
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Caravantes, J., Diaz–Toca, G.M., Gonzalez–Vega, L. (2023). Avoiding the General Position Condition When Computing the Topology of a Real Algebraic Plane Curve Defined Implicitly. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_46
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DOI: https://doi.org/10.1007/978-3-031-38271-0_46
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