This paper studies planar motion approximation problems from a computational geometric perspective and develop a computational geometric structure that can be used for mechanical motion synthesis. This allows for development of computational algorithms and software systems to support the mechanical design activity. The approach uses an orientable kinematic mapping to transform the mechanical design problem into a curve design problem in the space of the mapping. The curve design problem for synthesis of an analytic motion is carried out by Hermite interpolation. In case of a mechanical linkage, however, the Hermite interpolation is combined with a first-order curve fitting procedure for synthesizing the motion.