Visualizing Planar and Space Implicit Real Algebraic Curves with Singularities

TL;DR

This paper introduces a robust numerical method for visualizing implicit real algebraic curves in 2D and 3D, effectively handling singularities and higher-dimensional space curves through continuation and critical point techniques.

Contribution

The paper presents a novel visualization algorithm that accurately approximates algebraic curves near singularities and extends to higher dimensions, improving stability and topological analysis.

Findings

Effective visualization of algebraic curves with singularities

Stable polygonal approximation close to the actual curve

Enhanced algorithm with clustering and pseudo singular point handling

Abstract

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 $\epsilon$-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 $\mathbb{R}^2$, 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.