Research On The Structure Functions Of Ruled Surface

The ruled surface is a surface which is generated by the trace of a straight line,and it is widely used on Construction Engineering,CAD/CAM and Computer-based Animation Design.In previous studies,the ruled surface is expressed as a standard equation:the base curve is the striction line of the ruled surface and the direction line is an unit spherical curve with the arc-length parameter.we have defined the structure functions:λ(u),μ(u)and κg(u).Using one of the structure functions μ(u),we can classify the ruled surfaces as following:when μ(u)is zero,the surface is a developable ruled surface;when μ(u)is not zero,the surface is a non-developable ruled surface.Furthermore,when λ(u)is zero,the ruled surface is a binormal surface.Our work is mainly divided into two parts:In the first part,we study the structure functions of quadric surfaces which are the hyperbolic paraboloid,hyperboloid of the one sheet of revolution and hyperboloid of the one sheet of non-revolution.Firstly,we obtain the rectilinear generator and the directrix of corresponding ruled surface by considering the general parameter of quadric non-developable ruled surface as the function of arc-length parameter.Then we obtain the corresponding structure functions.In the second part,we discuss the ruled surface with particular striction line,and describe the characterization of the surfaces by using the structure functions.We can classify the nondevelopable ruled surfaces with constant λ(u)and κ(u)vanishes identically as the striction line are the spherical curve,the general helix and Bertrand curve,etc.respectively.