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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, itis 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 fornon-degenerate planar curves, equality of Euclidean signatures impliescongruence. We prove that while Hickman's claim holds for simple, closed curveswith simple signatures, it fails for curves with non-simple signatures. In thelater case, we associate a directed graph with the signature and show howvarious paths along the graph give rise to a family of non-congruent,non-degenerate curves with identical signatures. Using this additionalstructure, we formulate congruence criteria for non-degenerate, closed, simplecurves and show how the paths reflect the global and local symmetries of thecorresponding curve.

Non-congruent non-degenerate curves with identical signatures