Structure and characterization of ruled surfaces in Euclidean 3-space

Keywords: Ruled surface Structure function Pitch function Angle function of pitch Weingarten surface Binormal ruled surface a b s t r a c t In this paper, using the elementary method we study ruled surfaces, the simplest foliated submanifolds, in Euclidean 3-space. We define structure functions of the ruled surfaces, the invariants of non-developable ruled surfaces and discuss geometric properties and kine-matical characterizations of non-developable ruled surface in Euclidean 3-space. Müller [6] introduced the concepts of the pitch and angle of pitch of a closed ruled surface in Euclidean 3-space. In [4,5] the authors generalized these notions to pitch (density) function and angle (density) function of pitch (or according to the kine-matical meaning, self spinning density function) of any non-developable ruled surfaces in Euclidean 3-space and Minkowski 3-space. Some properties and applications of these notions are also given in [4,5]. For example, the B-scroll in Minkowski 3-space is characterized by that the pitch function of the ruled surface with lightlike ruling vanishes identically (Theorem 3.3 in [5]) ([5] for the concept of B-scrolls). In this paper we consider non-developable ruled surfaces in Euclidean 3-space. At first, we give the relations between the structure functions of the ruled surface and the curvature, torsion of the striction line of the ruled surface. Then we study normal ruled surface of the space curves and some properties of the non-developable ruled surfaces using the structure functions. As we know that, the tangent ruled surface of a space curve is developable ruled surface; the binormal ruled surface of a space curve is non pitched ruled surface. Finally, using the theory of limits, we define distance density function, translation density function and unit common perpendicular vector field of the non-developable ruled surface in Euclidean 3-space and give a new kinematical characterization of the non-developable ruled surface and also the relations with the structure functions in Euclidean 3-space. The ruled surfaces are the simplest foliated submanifolds. It is meaningful to generalized our methods to study other foliated submanifolds, for example, the surfaces foliated by circles. 2. Structure functions of ruled surfaces In this section we recall some concepts and results given in [4,5] (also in [1]). Some of them are closely related to the conclusions given in the next section.