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 isotopic correctness, with proven complexity bounds and a novel intersection detection test.

Contribution

It develops a new simple test for guaranteed correctness and extends subdivision algorithms to handle multiple curves with complexity analysis.

Findings

The algorithm guarantees isotopic approximation of multiple curves.

A new test ensures correct intersection detection.

Complexity bounds are established for the algorithm.

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.