Research And Application Of Rotation Minimizing Frame Along 3D Curves

The rotation minimizing frames of 3D curve have attracted much attention in research areas of computer graphics, computer animation and so on. They have important application in sweep surface modeling, three dimensional roaming and movement interpolation. As there is no explicit formulation for calculating rotation minimizing frames, the computation of rotation minimizing frames of 3D curve becomes a question which must be solved. At present, scholars have done much research for computing rotation minimizing adapted frame and rotation minimizing directed frame, and many kinds of effective geometry algorithms have been proposed.As implicit curve is another representation of curves, it has wide applications in practice. Starting from the relationship between the angular velocity and linear velocity of movement vector in the 3D space, this paper focuses on computation of rotation minimizing frames based on previous research. First, the ordinary differential equations for vectors of the rotation minimizing adapted frames along 3D implicit curve are deduced. Secondly, the first and second derivatives of axial vectors with respect to the arc length parameter of 3D implicit curve are derived. Then a second-order iterative algorithm based on Taylor approximation for computing rotation minimizing adapted frame is presented. On the other hand, as the existing algorithm for calculating rotation minimizing directed frame has limitation, we derive the ordinary differential equations for this frame, and then solve it by a classical fourth order Runge-Kutta method.Experimental results indicate our algorithms have good precision. The algorithm for computing the rotation minimizing directed frame overcomes the insufficiency of existing approaches. We further apply the algorithms to sweep surface modeling and camera orientation controlling and gain satisfactory results.