Assessing Isometric Folding in Curved-Crease Origami: A Discretized Algorithm with Outlook Toward Stretch Localization

Curved-crease origami structures often exhibit geometric frustration: under certain boundary conditions, folding without in-plane stretching is impossible. We present a discretized geometric algorithm that determines whether a given curved-crease system can fold isometrically. Furthermore, our algorithm can indicate how the crease's geometry may be modified to approach stretch-free folding. This framework provides a concrete tool for predicting and tuning multistability in non-Euclidean origami. We demonstrate its use on a discretized parabolic reflector that folds between two stable states, revealing how curvature and boundary constraints govern mechanical compatibility. When facet stretching is unavoidable, we propose a complementary strategy to localize strain along the creases and visualize the resulting energy landscape using a variational autoencoder (VAE). The VAE enables intuitive design by identifying the locations and shapes of stable states in high-dimensional configuration space. This combination of geometric and data-driven approaches offers a new pathway for designing multistable, stretch-tolerant origami inspired by natural systems such as the deployable earwig wing.