Certified simultaneous isotopic approximation of curves via subdivision

TL;DR

This paper introduces a certified subdivision-based algorithm for accurately approximating multiple curves in the plane while ensuring topological correctness and providing complexity analyses.

Contribution

It extends existing curve approximation methods by introducing a new test for intersection correctness and analyzes the algorithm's complexity.

Findings

Guarantees global correctness of curve approximations.

Provides complexity bounds for the subdivision process.

Ensures accurate intersection detection among multiple curves.

Abstract

We present a certified algorithm based on subdivision for computing an isotopic approximation to any number of curves in the plane. Our algorithm is based on the certified curve approximation algorithm of Plantinga and Vegter. The main challenge in this algorithm is to correctly and efficiently identify and isolate all intersections between the curves. To overcome this challenge, we introduce a new and simple test that guarantees the global correctness of our output. A main step in our algorithm for approximating any number of curves is to correctly approximate a pair of curves. In addition to developing the details of this special case, we provide complexity analyses for both the number of steps and the bit-complexity of this algorithm using both worst-case bounds as well as those based on continuous amortization.