We prove the Jordan curve theorem by generalizing the sweepline algorithm for trapezoidal decomposition of a polygon. Our proof uses Zorn's lemma (or, equivalently the axiom of choice). Though several proofs have been given for the Jordan curve theorem by various authors, ours is the {\bf first algorithmic proof} of Jordan curve theorem using computational geometry. Further, with some preparation, the proof can be taught as part of an undergraduate discrete mathematics course, where till now the proof of this theorem was considered inaccessible.

翻译：我们通过推广用于多边形梯形分解的扫描线算法，证明了若尔当曲线定理。本证明使用了佐恩引理（等价于选择公理）。尽管已有不同作者给出了若尔当曲线定理的多种证明，但我们的证明是首个利用计算几何的**算法化证明**。此外，经过适当准备，本证明可作为本科生离散数学课程的教学内容——而此前该定理的证明被认为难以在此层次教授。