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.

翻译：本文提出一种基于细分法的认证算法，用于计算平面内任意数量曲线的保同伦逼近。该算法建立在Plantinga和Vegter的认证曲线逼近算法基础之上。算法的主要挑战在于如何正确且高效地识别并分离所有曲线间的交点。为克服这一挑战，我们引入了一种新颖而简洁的判定方法，确保输出结果的全局正确性。逼近任意数量曲线的主要步骤是正确处理两条曲线的逼近问题。除详细阐述该特殊情形的实现细节外，我们通过最坏情况界与连续摊销分析两种方法，对该算法的迭代步数复杂度与比特复杂度进行了理论分析。