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arXiv · 1912.09597

Non-congruent non-degenerate curves with identical signatures

Eric Geiger·Irina A. Kogan
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Abstract

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 later 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.

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Eric Geiger, Irina A. Kogan. 2019-12-20. Non-congruent non-degenerate curves with identical signatures. https://doi.org/10.1007/s10851-020-01015-x

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