Certified algebraic curve projections by path tracking

2025-02-07

AI Summary

This paper addresses the problem of computing topologically faithful projections of smooth algebraic curves in ℝⁿ (implicitly defined) onto ℝ², aiming to produce planar approximations that rigorously preserve crossing structure and homeomorphism type. The proposed method integrates certified path tracking with interval arithmetic: first, certified numerical continuation extracts critical topological event points—such as critical points and self-intersections—along the curve; second, interval arithmetic is employed to verify local injectivity of the projection map and global topological consistency; finally, a provably correct planar embedding is reconstructed. This work introduces the first application of certified path tracking to projection topology verification, yielding a mathematically guaranteed homeomorphic approximation. Experiments demonstrate robustness across arbitrary-dimensional algebraic curves, producing certified topologically equivalent planar projections. The approach establishes a rigorous foundation for geometric modeling and visualization of high-dimensional algebraic curves.

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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.

Problem

• Computing isotopic approximations for curve projections
• Ensuring topological correctness of crossings
• Combining certified path tracking and interval arithmetic

Innovation

• Certified path tracking technique
• Interval arithmetic refinement process
• Topological correctness guarantee method