The kinematic properties and synthesis of single and multiple degrees-of-freedom pencil trajectories are investigated in this research project. Whereas a line trajectory is one in which the tracing element follows a line, in the case of a pencil trajectory, both the line and the position of a point on the line are included. It is therefore feasible to control the movement of the tip of the pencil and the orientation of the pencil. For a single degree-of-motion a pencil trajectory may be regarded as a point trajectory on a ruled surface. Typical applications of such motions requiring control of a pencil include drilling, screwing, welding, cutting, drawing, and laser or water jet positioning. In the area of design and manufacturing, results from this research provide an integrated treatment of both the direction and position of a line segment, there by eliminating the unnecessary constraint of a rigid body motion. This leads to a considerable simplification for the pencil trajectory synthesis and planning. The simplification is especially valuable when the number of degrees of freedom is limited to achieve high speed, high precision, easy of operation, and low cost.***//
