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In this paper we consider developable surfaces which are isometric to planardomains and which are piecewise differentiable, exhibiting folds along curves.The paper revolves around the longstanding problem of existence of theso-called folded annulus with multiple creases, which we partially settle bybuilding upon a deeper understanding of how a curved fold propagates toadditional prescribed foldlines. After recalling some crucial properties ofdevelopables, we describe the local behaviour of curved folding employingnormal curvature and relative torsion as parameters and then compute the verygeneral relation between such geometric descriptors at consecutive folds,obtaining novel formulae enjoying a nice degree of symmetry. We make use ofthese formulae to prove that any proper fold can be propagated to an arbitraryfinite number of rescaled copies of the first foldline and to give reasons whyproblems involving infinitely many foldlines are harder to solve.

Propagation of curved folding: The folded annulus with multiple creases exists