Let $\gamma$ be a generic closed curve in the plane. Samuel Blank, in his 1967 Ph.D. thesis, determined if $\gamma$ is self-overlapping by geometrically constructing a combinatorial word from $\gamma$. More recently, Zipei Nie, in an unpublished manuscript, computed the minimum homotopy area of $\gamma$ by constructing a combinatorial word algebraically. We provide a unified framework for working with both words and determine the settings under which Blank's word and Nie's word are equivalent. Using this equivalence, we give a new geometric proof for the correctness of Nie's algorithm. Unlike previous work, our proof is constructive which allows us to naturally compute the actual homotopy that realizes the minimum area. Furthermore, we contribute to the theory of self-overlapping curves by providing the first polynomial-time algorithm to compute a self-overlapping decomposition of any closed curve $\gamma$ with minimum area.

翻译：令 $\gamma$ 为平面上的一般闭曲线。Samuel Blank 在其 1967 年的博士论文中，通过从 $\gamma$ 几何构造一个组合词语，确定了 $\gamma$ 是否自重叠。最近，Zipei Nie 在一份未发表的手稿中，通过代数方式构造一个组合词语，计算了 $\gamma$ 的最小同伦面积。我们提供了一个统一框架来处理这两个词语，并确定了 Blank 词语与 Nie 词语等价的条件。利用此等价性，我们给出了 Nie 算法正确性的新几何证明。与以往工作不同，我们的证明是构造性的，从而能够自然地计算出实现最小面积的实际同伦。此外，我们通过提出首个多项式时间算法来计算任意闭曲线 $\gamma$ 的最小面积自重叠分解，为自重叠曲线理论做出了贡献。