Geodesic
Piecewise Smooth Developable Surfaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 227-250

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A. V. Pogorelov introduced developable surfaces with regularity (twice differentiability) violated along separate lines. In particular, the surface may not be smooth at all points of these lines (which form edges in this case). It is assumed that each point of the surface under consideration that belongs to a curvilinear edge (as well as any other interior point of this surface) has a neighborhood isometric to a Euclidean disk. In this paper we study the behavior of a developable surface near its curvilinear edge. It is proved that if two smooth pieces of a developable surface are adjacent along a curvilinear edge, then the spatial location of one of them in $\mathbb R^3$ is uniquely determined by that of the other. 
@article{TM_2008_263_a15,
     author = {M. I. Shtogrin},
     title = {Piecewise {Smooth} {Developable} {Surfaces}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {227--250},
     publisher = {mathdoc},
     volume = {263},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2008_263_a15/}
}
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M. I. Shtogrin. Piecewise Smooth Developable Surfaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 227-250. http://geodesic.mathdoc.fr/item/TM_2008_263_a15/