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.

中文翻译：

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可视化具有奇异性的平面和空间隐式实数代数曲线

本文提出了一种基于数值连续和临界点方法可视化2-D或3-D环境空间中边界框内隐式实数代数曲线的新方法。基础技术还可以用于跟踪高维空间中的空间曲线。由于曲线的奇异点附近的拓扑在数值上不稳定，因此作者仅跟踪曲线在奇异点邻域之外，并简单地用一个点替换每个邻域，从而产生与曲线e接近的多边形近似值。 。这种近似是更稳定的，用于定义在ℝ曲线的投影的互补的数值连通²，这对于求解双参数多项式系统等应用很重要。该算法从计算曲线的三种关键点开始，即曲线与以奇点为中心的小球的交点，曲线的每个连接分量的规则临界点以及曲线与给定点的交点边界框。然后，它按照上述三种类型的点并按其顺序跟踪曲线。通过一些优化，例如对自然簇中的奇异点进行分组，通过尝试-恢复策略跟踪曲线以及处理“伪奇异点”，进一步优化了该基本方案。大量示例说明了该算法的有效性。该手稿扩展了CASC 2018中出现的拟议初步结果。