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We derive new algebraic equations for the folding angle relationships incompletely general degree-four rigid-foldable origami vertices, including bothEuclidean (developable) and non-Euclidean cases. These equations in turn leadto novel, elegant equations for the general developable degree-four case. Wecompare our equations to previous results in the literature and provide twoexamples of how the equations can be used: In analyzing a family of squaretwist pouches with discrete configuration spaces, and for proving that a newfolding table design made with hyperbolic vertices has a single folding mode.

Explicit kinematic equations for degree-4 rigid origami vertices, Euclidean and non-Euclidean