A Continuation Method for Visualizing Planar Real Algebraic Curves with Singularities

We present a new method for visualizing planar real algebraic curves inside a bounding box based on numerical continuation and critical point methods. Since the topology of the curve near a singular point is not numerically stable, we trace the curve only outside neighborhoods of singular points and replace each neighborhood simply by a point, which produces a polygonal approximation that is \(\epsilon \)-close to the curve. Such an approximation is more stable for defining the numerical connectedness of the complement of the curve, which is important for applications such as solving bi-parametric polynomial systems.

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