Certified algebraic curve projections by path tracking

Abstract

We present a certified algorithm that takes a smooth algebraic curve in ${\mathbb{R}}^n$ and computes an isotopic approximation for a generic projection of the curve into ${\mathbb{R}}^2$. Our algorithm is designed for curves given implicitly by the zeros of $n-1$ polynomials, but it can be partially extended to parametrically defined curves. The main challenge in correctly computing the projection is to guarantee the topological correctness of crossings in the projection. Our approach combines certified path tracking and interval arithmetic in a two-step procedure: first, we construct an approximation to the curve in ${\mathbb{R}}^n$, and, second, we refine the approximation until the topological correctness of the projection can be guaranteed. We provide a proof-of-concept implementation illustrating the algorithm.

23 pages, 7 figures. Accepted for the Proceedings of ISSAC 2025