Authors’ abstract: “We consider a developable surface tangent to a surface along a curve on the surface. We call it an osculating developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface which characterize these singularities. As a by-product, we show that a curve is a contour generator with respect to an orthogonal projection or a central projection if and only if one of these invariants constantly equal to zero.”
The paper is organized into six sections dealing with the following aspects: basic concepts, osculating developable surfaces, support functions, invariants of curves on surfaces, curves on special surfaces (curves on developable surfaces, curves on the unit sphere, curves on the graph of a function). Other papers by the first author directly connected to this topic are [Geometry of ruled surfaces, in: Applicable Mathematics in the Golden Age, Narosa Publishing House, New Delhi, 305–338 (2003)] (with N. Takeuchi ) and [S. Izumiya et al., J. Math. Soc. Japan 62, No. 3, 789–849 (2010; Zbl 1205.53065)] (with K. Saji and M. Takahashi).