The Moving Frame Based Study Of The Property And Application Of Circular Surface

In this dissertation, the geometrical characteristics and properties of circular surface are studied by means of differential geometry. The Frenet frame of circular surface is built to derive the geometrical invariants of circular surfaces, based on which some degenerate circular surfaces and some constraint circular surfaces for spatial mechanism are analyzed.Firstly, the vector equation of circular surface is obtained through the central curve and the moving frame which is based on the unit normal vector of the circle plane and the arc length of its spherical image curve. Then the relation formula of the normal curvature, geodesic curvature and geodesic torsion of the circle generatrix is given. Through the first and second fundamental parameters, the properties of the normal fixed radius circular surface are proposed. The solving method of determination of the principle curvature and principle direction is discussed.Secondly, with the combination of exterior differential form and moving frame, the Frenet frame is built and it's proved that the origin trajectory which is called spine curve of the frame has the same geometric property as the striction curve of ruled surface. With the spine curve as directrix, the standard vector equation of circular surface is given.Finally, the geometrical properties of the degenerate circular surface are discussed when the central curve of circular surface becomes a circle, cylindrical spiral, spherical curve or a point. And also, the sufficient conditions are proved for a general circular surface to be one of above four types of circular surfaces. To the constraint circular surface for spatial mechanisms, the building process of the Frenet frame is proposed and the invariants are discussed. It's essential to lay groundwork of kinematic differential geometry of spatial mechanisms.