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Comparing the shapes of two curves requires more than placing their points side by side. The authors study elastic registration, in which reparametrization and rotation are used to align curves before a shape distance is computed.
They develop a dynamic programming procedure for optimal reparametrizations and an algebraic treatment of the Kabsch–Umeyama rotation algorithm. The report also considers efficient computations across starting points, including the use of fast Fourier transforms for closed curves. Definitions, derivations, algorithms and numerical examples give the complete mathematical development.
Javier Bernal, Jim Lawrence, Günay Doğan and Charles Hagwood (2021). On Computing Elastic Shape Distances between Curves in d-dimensional Space. National Institute of Standards and Technology. https://doi.org/10.6028/NIST.TN.2164. Republished courtesy of the National Institute of Standards and Technology.