Characteristics Of Positive Cone Curves And Application Of Structure Function

Positive cone plays an important role in geometry study and engineering applications, so it is worth for us to study curves on it, however, the traditional methods of researching on positive cone has a certain limitation.In this thesis, we define the structure function to show the expressions of curvature and torsion of the curves with a special parameter which is called "the cone parameter", trying to find some conclusions by analyzing them. After that, we prove that the plane curves on the positive cone are all special kind with orthogonal transformation and the classification of the plane curves is given. Some differential equations are obtained from the expressions of curvature and torsion of the certain curves, and the particular solutions to these differential equations are given by using corresponding properties.In the following chapter the is introduced and we show the expressions of curvature and torsion of the positive cone curves by it, then the relation between the translation of polar coordinates and the structure function is uncovered. By comparison, describing the curves with polar coordinates is simple in form while inconvenient when dealing with realistic situation.