We consider the problem of classifying curves when they are observed only partially on their parameter domains. We propose computational methods for (i) completion of partially observed curves; (ii) assessment of completion variability through a nonparametric multiple imputation procedure; (iii) development of nearest neighbor classifiers compatible with the completion techniques. Our contributions are founded on exploiting the geometric notion of shape of a curve, defined as those aspects of a curve that remain unchanged under translations, rotations and reparameterizations. Explicit incorporation of shape information into the computational methods plays the dual role of limiting the set of all possible completions of a curve to those with similar shape while simultaneously enabling more efficient use of training data in the classifier through shape-informed neighborhoods. Our methods are then used for taxonomic classification of partially observed curves arising from images of fossilized Bovidae teeth, obtained from a novel anthropological application concerning paleoenvironmental reconstruction.