Efficient topology determination of implicitly defined algebraic plane curves

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Abstract

This paper is devoted to present a new algorithm computing in a very efficient way the topology of a real algebraic plane curve defined implicitly. This algorithm proceeds in a seminumerical way by performing a symbolic preprocessing which allows later to accomplish the numerical computations in a very accurate way.

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    Citation Excerpt :

    In this section, we study the cutcurve of the two quadrics whose intersection curve is to be computed; this is achieved by computing its topology. The usual method of computing the topology of a real algebraic plane curve defined implicitly requires computing its critical points (those that are singular or with a vertical tangent), regular points sharing the same projection as one of the critical points and the number of branches connecting all these points (connecting branches and points will be particularly easy in our case, because we are working with a very special quartic): for details see [33,34] and Section 4.7 (especially Fig. 3). In our case, because we are working with the cutcurve, we need to restrict our attention to the admissible region; this is why we will also pay special attention to the intersection points between the cutcurve and the silhouette curves.

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Partially supported by DGES PB98-0713-C02-02 and by FEDER project 1FD-97-0409.

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