This paper presents a new approach for determining three-dimensional global displacement (for arbitrarily sized deformation) of thin rod or tetherlike structures from a limited set of scalar strain measurements. The approach is rooted in Cosserat rod theory with a material-adapted reference frame and a localized linearization approach that facilitates an exact local basis function set for the displacement along with the material frame. The solution set is shown to be robust to potential singularities from vanishing bending and twisting angle derivatives and from vanishing measured strain. Validation of the approach is performed through a comparison with both finite element simulations and an experiment, with average root mean square reconstruction error of 0.01%–1% of the total length, for reasonable sensor counts. An analysis of error due to extraneous noise sources and boundary condition uncertainty shows how the error scales with those effects. The algorithm involves relatively simple operations, the most complex of which is square matrix inversion, lending itself to potential low-power embeddable solutions for applications requiring shape reconstruction.