Ruled Surface And Spherical Curve

The integral invariant is very important in the study of ruled surface. But it is well known that the integral invariant (the pitch and the angle of pitch) of the closed ruled surface currently. In this paper, we extend the definitions of the integral invariant of the closed ruled surface to the general ruled surface. We study the integral invariant of the ruled surface and the instant rotary axis surface of the curves on the ruled surface.In this paper, We start with setting up the Frenet frame of the curves on the ruled surface, and define two functions:λ(u) and μ(u). λ(u) affects the bended degree of the ruled surface. μ(u) is showed as the definition of the pitch of the ruled surface. We study the relationship between the quantities of the directrix(line of striction) a(u) and the quantities of the spherical curve b(u), based on the Frenet frame.Then we study the fundamental quantities of the instant rotary axis surface of the spherical curve b(u): r1(u,w)=b(u)+wΩ(u) and the instant rotary axis surface of the directrix a(u): r2(u, w)=a(u)+wΩ(u). We count their Gauss curvature and mean curvature, and obtain some results.