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The existence of osculating planes is established for a large class of differentiable arcs in , called "coherent"; all analytic arcs, including the singular ones, belong to this family. On a coherent arc, osculating planes and spheres exist at any point and vary differentiably; Frenet formulas and curvatures are reformulated in order to generalize the classical ones. Coherent arcs form an open set in the space of all arcs, with infinite codimensional complement.
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