Non-congruent non-degenerate curves with identical signatures
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.
Keywords
Cite
@article{arxiv.1912.09597,
title = {Non-congruent non-degenerate curves with identical signatures},
author = {Eric Geiger and Irina A. Kogan},
journal= {arXiv preprint arXiv:1912.09597},
year = {2021}
}
Comments
33 pages, 22 figures. Page 20: In the proof of Corollary 31 the notation for the length, $L_W$, of a reconstructed curve $\Gamma_W$ is introduced and defined. Page 23: The upper bound on the integral in equation (35) is updated to use $L_W$ instead of $L$ and the definition of $L_W$ is referred to. Page 23: The assumption "$m$ and $\xi$ are relatively prime" is added to Proposition 36