A new method, for the investigation of manipulator workspace, based on polynomial displacement equations and their discriminants is proposed in this paper. The approach followed enables one to obtain (a) analytical expressions describing the workspace boundary surfaces in Cartesian coordinates, (b) the distribution of the number of ways to position the hand inside the workspace, and (c) the conditions on kinematic parameters when the motion of the hand degenerates. Joint limitations are incorporated in the method. In the present paper and a companion paper [17], the method is rigorously applied to eight types of manipulators with various combinations of revolute and prismatic pairs having the last three revolute axes intersect orthogonally at at point. A numerical example is presented for illustration.