2019/12/16 by Changbo Chen, Wenyuan Wu, Chen, Changbo +3
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Commutative Algebra and Its Applications #Computational Geometry (cs.CG) #FOS: Computer and information sciences #Polynomial and algebraic computation #Symbolic Computation (cs.SC)
paper · pdf · doi:10.48550/arxiv.1912.07507
openalex publication_date 2019/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present 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, we trace only the curve outside neighborhoods of singular points and replace each neighborhood simply by a point, which produces a polygonal approximation that is ε-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 our preliminary results that appeared in CASC 2018.