A continuation algorithm for planar implicit curves with singularities

• An algorithm to numerically resolve critical points of implicitly defined functions and curves.
• An algorithm to find singularities of implicitly defined functions and curves.
• An algorithm capable of detecting multiple components of implicit curves.
• An algorithm capable of numerically determining the topology of vector fields.

Continuation algorithms usually behave badly near to critical points of implicitly defined curves in R2R2, i.e., points at which at least one of the partial derivatives vanishes. Critical points include turning points, self-intersections, and isolated points. Another problem with this family of algorithms is their inability to render curves with multiple components because that requires finding first a seed point on each of them. This paper details an algorithm that resolves these two major problems in an elegant manner. In fact, it allows us not only to march along a curve even in the presence of critical points, but also to detect and render curves with multiple components using the theory of critical points.

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