Tangential developable

Abstract: We define new special curves in Euclidean 3-space which we call slant helices and conical geodesic curves Those notions are generalizations of the notion of cylindrical helices One of the results in this paper gives a classification of special developable surfaces under the condition of the existence of such a special curve as a geodesic As a result, we consider geometric invariants of space curves By using these invariants, we can estimate the order of contact with those special curves for general space curves All arguments in this paper are straight forward and classical However, there have been no papers which have investigated slant helices and conical geodesic curves so far as we know

Abstract: A generalization of the theory of Bertrand curves is presented for ruled and developable surfaces based on line geometry. Using lines instead of points as the geometric building blocks of space, two ruled surfaces which are offset in the sense of Bertrand are defined. It is shown that, in general, every ruled surface can have a double infinity of Bertrand offsets; but for a developable ruled surface to have a developable Bertrand offset, a linear equation should be satisfied between the curvature and torsion of its edge of regression. In addition, it is shown that the developable offsets of a developable surface are parallel offsets. The results, in addition to being of theoretical interest, have applications in geometric modelling and the manufacturing of products.