Curves On Hyperboloid Of Two Sheets

Hyperboloid plays a important part in engineering applications, there are two kinds of hyperboloid:hyperboloid of one sheet and hyperboloid of two sheets. Hyperboloid of one sheet is ruled surface and studied in many papers, however, relevant studies on hyperboloid of two sheets are limited.In this thesis we discuss the curves on positive hyperboloid of two sheets, by introducing the concept of structure function, we try to express the curvature、the torsion and other geometric quantities of the curves on hyperboloid of two sheets with structure function. Then we are going to find curves on positive hyperboloid of two sheets which have special geometric characteristics. Through the discussions, the espressions of the curvature and torsion of some plane curves are given.On this basis, we discuss two kinds of curves on hyperboloid of two sheets. First of all, we discuss the existence of the general helix, then the equations and the espressions of the curvature and torsion of the general helixes are given. And then we study the existence of rectifying curves and obtain the equation of the curves that put its position vector as immediately rotation axis.