tailieunhanh - New special curves and developable surfaces

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. | Turk J Math 28 (2004) , 153 – 163. ¨ ITAK ˙ c TUB New Special Curves and Developable Surfaces Shyuichi Izumiya, Nobuko Takeuchi 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. Key Words: Helix, Darboux vector, developable surfaces, singularities 1. Introduction In [3] we have studied singularities of the rectifying developable (surface) of a space curve. The rectifying developable is defined to be the envelope of the family of rectifying planes along the curve. We have also studied the Darboux developable of a space curve whose singularities are given by the locus of the end points of modified Darboux vectors of the curve [3, 5, 6]. In this paper we define the notion of slant helices and conical geodesic curves which are generalizations of the notion of helices (cf., §2). We introduce the notion of the tangential Darboux developable of a space curve which is defined by the Darboux developable of the tangent indicatrix of the space curve (cf., §3). We study singularities of the tangential Darboux developable of a space curve as an application of the classification theorem of 153 IZUMIYA, TAKEUCHI developable surfaces in [6]. We find a geometric invariant of a space curve which is deeply related to the singularities of the tangential Darboux developable of the original curve. In §2 we describe basic .