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Flat surfaces along cuspidal edges. (English) Zbl 1376.57034

The authors consider developable surfaces along the singular curve of a cuspidal edge surface in the Euclidean \(3\)-space. Since a cuspidal edge surface has the normal direction at any point (even at a singular point), they focus on two typical developable surfaces along the cuspidal edge. The first one is a developable surface which is tangent to the cuspidal edge surface and the second is normal to the cuspidal edge surface. The authors investigate the singularities of these developable surfaces along the cuspidal edge and introduce new invariants for the cuspidal edge.
The paper is organized into seven sections dealing with the following aspects: Introduction, cuspidal edges, Darboux frames along cuspidal edges, developable surfaces and generalizations of helices, developable surfaces along cuspidal edges (osculating developable surfaces along cuspidal edges, normal developable surfaces along cuspidal edges, planer cuspidal edges, normalized derivate director curves and derivate striction curves), special cuspidal edges (contour edges, isophotic edges, general order sloped edges), curves on regular surfaces and relationships with cuspidal edges. At the end of the paper a useful Appendix about support functions, and a relevant list of references are given.

MSC:

57R45 Singularities of differentiable mappings in differential topology
58K99 Theory of singularities and catastrophe theory
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