Defining a “norm” that quantifies a measure of “distance” for rigid body displacement in plane or space, is essential to many problems in kinematics, dynamics and control of mechanical and manufacturing systems. The norm is an abstraction of our usual concept of the length. The mapping between different generalized coordinate systems used in the analysis or control of systems may also be derived from this norm. All such norms currently used in different analysis are highly sensitive to 1) coordinate systems defined, and 2) metrics defined. A new class of norms is proposed to quantify the distance in the space. These norms are independent of the coordinate systems or metrics used. Its applications indesign, precision manipulation and control, computational kinematics and dynamics, assembly and part mating in manufacturing, space kinematics and dynamics, shape optimization and motion planning are explored.