Non-congruent non-degenerate curves with identical signatures

12/20/2019

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by   Eric Geiger, et al.

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We construct examples of non-congruent, non-degenerate simple planar closed curves with identical Euclidean signatures, thus disproving a claim made in Hickman (J. Math Imaging Vis. 43:206-213, 2012) that all such curves must be congruent. Our examples include closed C^∞ curves of the same length and the same symmetry group. We show a general mechanism for constructing such examples by exploiting the self-intersection points of the signature. We state an updated congruence criterion for simple closed non-degenerate curves and confirm that for curves with simple signatures the claim made by Hickman holds.