Certified algebraic curve projections by path tracking

TL;DR

This paper introduces a certified algorithm that accurately computes topologically correct projections of smooth algebraic curves from higher dimensions into 2D, ensuring the preservation of crossings and topology.

Contribution

The paper presents a novel certified algorithm combining path tracking and interval arithmetic to produce topologically correct projections of algebraic curves.

Findings

Algorithm guarantees topological correctness of projections.

Implementation demonstrates practical viability.

Applicable to implicit algebraic curves, extendable to parametric cases.

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.