While the equality of differential signatures (Calabi et al., Int. J. Comput. Vis. 26: 107–135, 1998) is known to be a necessary condition for congruence, it is not sufficient (Musso and Nicolodi, J. Math Imaging Vis. 35: 68–85, 2009). Hickman (J. Math Imaging Vis. 43: 206–213, 2012, Theorem 2) claimed that for non-degenerate planar curves, equality of Euclidean signatures implies congruence. We prove that while Hickman’s claim holds for simple, closed curves with simple signatures, it fails for curves with non-simple signatures. In the latter case, we associate a directed graph with the signature and show how various paths along the graph give rise to a family of non-congruent, non-degenerate curves with identical signatures. Using this additional structure, we formulate congruence criteria for non-degenerate, closed, simple curves and show how the paths reflect the global and local symmetries of the corresponding curve.

虽然已知差异签名的相等性（Calabi等人，Int。J. Comput。Vis。26：107-135，1998）是一致的必要条件，但这还不够（Musso和Nicolodi，J。Math。影像视觉35：68-85，2009年）。希克曼（J. Math Imaging Vis。43：206–213，2012，定理2）声称，对于非简并的平面曲线，欧几里得签名的相等性意味着同余。我们证明，尽管希克曼的主张适用于具有简单签名的简单闭合曲线，但对于具有非简单签名的曲线却不适用。在后一种情况下，我们将有向图与签名相关联，并显示沿图的各种路径如何产生具有相同签名的一系列非一致，非简并曲线。使用此附加结构，我们为非退化，封闭，