A Continuation Method for Visualizing Planar Real Algebraic Curves with Singularities
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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.
The algorithm starts by computing three types of key points of the curve, namely the intersection of the curve with small circles centered at singular points, regular critical points of every connected component of the curve, as well as intersection points of the curve with the given bounding box. It then traces the curve starting with and in the order of the above three types of points. This basic scheme is further enhanced by several optimizations, such as grouping singular points in natural clusters and tracing the curve by a try-and-resume strategy. The effectiveness of the algorithm is illustrated by numerous examples.
The authors would like to thank Chee K. Yap and the reviewers, in particular Reviewer 3, for valuable suggestions. This work is partially supported by the projects NSFC (11471307, 11671377, 61572024), and the Key Research Program of Frontier Sciences of CAS (QYZDB-SSW-SYS026).
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