Moving coordinate frames for representation and visualization in four dimensions

In this paper we generalize the method of sweeping, which may be used for both object representation and object visualization, to objects in E 4. The generalized sweeping method involves a trajectory in E 4 , and a three-dimensional cross-section orthogonal to the trajectory. Three alternatives are considered for specifying the relationship between the trajectory and the cross-section, namely the Frenet frame, modiied Frenet frames, and the standard viewing-reference coordinate (VRC) frame, where each of these is generalized to the case of four dimensions. We have implemented the method using the four-dimensional VRC frame, and the method is illustrated here by means of two examples involving visualization of objects embedded in E 4 .