New special curves and developable surfaces. (English) Zbl 1081.53003

Authors’ abstract: We define new special curves in Euclidean 3-space which we call slant helices and conical geodesic curves. These 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 these 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 as far as we know.


53A04 Curves in Euclidean and related spaces
53A05 Surfaces in Euclidean and related spaces