Certified Rational Parametric Approximation of Real Algebraic Space Curves with Local Generic Position Method

TL;DR

This paper presents a certified method for approximating algebraic space curves with rational parametric curves, ensuring topological correctness and bounded Hausdorff distance by extending the local generic position technique.

Contribution

It introduces a novel algorithm that extends the local generic position method to space curves, enabling simultaneous topology computation and approximation with explicit error bounds.

Findings

The algorithm guarantees topology preservation of the approximation.

It provides explicit Hausdorff distance bounds based on plane curve approximations.

Effective examples demonstrate the method's practicality.

Abstract

In this paper, an algorithm to compute a certified $G^1$ rational parametric approximation for algebraic space curves is given by extending the local generic position method for solving zero dimensional polynomial equation systems to the case of dimension one. By certified, we mean the approximation curve and the original curve have the same topology and their Hausdauff distance is smaller than a given precision. Thus, the method also gives a new algorithm to compute the topology for space algebraic curves. The main advantage of the algorithm, inhering from the local generic method, is that topology computation and approximation for a space curve is directly reduced to the same tasks for two plane curves. In particular, the error bound of the approximation space curve is obtained from the error bounds of the approximation plane curves explicitly. Nontrivial examples are used to show the effectivity of the method.