For two surfaces glued along a curve, the paper classifies the gluing as cylindrical, conical, cuspidal-edgy, or swallowtailed according to explicit conditions on the frame invariants of the two surfaces and the angle between their normals.
Ishikawa, Singularities of flat extensions from generic surfaces with bound- aries.Differential Geometry and its Applications,28(2010), no
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Geometry on the Gluing Locus of Two Surfaces
For two surfaces glued along a curve, the paper classifies the gluing as cylindrical, conical, cuspidal-edgy, or swallowtailed according to explicit conditions on the frame invariants of the two surfaces and the angle between their normals.