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Visualizing Planar and Space Implicit Real Algebraic Curves with Singularities

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Abstract

This paper presents a new method for visualizing implicit real algebraic curves inside a bounding box in the 2-D or 3-D ambient space based on numerical continuation and critical point methods. The underlying techniques work also for tracing space curve in higher-dimensional space. Since the topology of a curve near a singular point of it is not numerically stable, the authors trace only the curve outside neighborhoods of singular points and replace each neighborhood simply by a point, which produces a polygonal approximation that is e-close to the curve. Such an approximation is more stable for defining the numerical connectedness of the complement of the projection of the curve in ℝ2, 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 spheres centered at singular points, regular critical points of every connected components 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, tracing the curve by a try-and-resume strategy and handling “pseudo singular points”. The effectiveness of the algorithm is illustrated by numerous examples. This manuscript extends the proposed preliminary results that appeared in CASC 2018.

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Acknowledgements

The authors would like to thank Chee K. Yap for many valuable suggestions, Michael Monagan for his advice on usage of Plots:-implicitplot, and anonymous referees for helpful suggestions and comments.

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Authors and Affiliations

  1. Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing, 400714, China

    Changbo Chen, Wenyuan Wu & Yong Feng

  2. University of Chinese Academy of Sciences, Beijing, 100049, China

    Changbo Chen, Wenyuan Wu & Yong Feng

Authors
  1. Changbo Chen
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  2. Wenyuan Wu
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  3. Yong Feng
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Corresponding author

Correspondence to Wenyuan Wu.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant Nos. 11771421, 11671377, 61572024, CAS “Light of West China” Program, cstc2018jcyj-yszxX0002 of Chongqing and the Key Research Program of Frontier Sciences of CAS under Grant No. QYZDB-SSW-SYS026.

This paper was recommended for publication by Editor LI Hongbo.

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Chen, C., Wu, W. & Feng, Y. Visualizing Planar and Space Implicit Real Algebraic Curves with Singularities. J Syst Sci Complex 33, 1252–1274 (2020). https://doi.org/10.1007/s11424-020-8380-0

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  • DOI: https://doi.org/10.1007/s11424-020-8380-0

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